A    STUDY     OF    THE    CIRCULAR-ARC 

BOW-GIRDER 


THE    CIRCULAR- ARC 
BOW-GIRDER 


BY 

A,  H.   GIBSON 

D.SC.,  ASSOC.MEM.INST.C.K.,  M.I.MECH.E. 
Professor  of  Engineering  in  the  University  of  Si.  Andrews,   University  College,  Dundee 

AND 

E.  G.  RITCHIE 

B.SC. 
,/lssistant  Lecturer  in  Engineering,   University  College,  Dundee 


NEW  YORK 

D.   VAN   NOSTRAND   COMPANY 
25  PARK  PLAGE 

1915 


Printed  in  Great  Britain 


PREFACE 


THK  problem  of  the  state  of  equilibrium  and  of  stress  of  the  circular-arc  bow-girder, 
i.e.,  the  girder  forming  a  circular  arc  in  plan  such  as  is  often  used  to  support  the 
balcony  of  a  theatre,  is  one  affording  some  difficulties  of  solution.  These  arise  mainly 
from  the  fact  that  in  addition  to  the  bending  moments  and  reactions  involved  in  the 
case  of  the  straight  encastre  girder,  twisting  moments  are  called  into  play  at  each 
section  and  at  the  ends  of  the  bow-girder,  and  these  moments  affect  very  considerably 
the  state  of  equilibrium  of  the  girder. 

The  general  problem  was  solved  in  a  paper  read  before  the  Eoyal  Society  of  Edin- 
burgh by  Professor  Gibson  in  1912,  and  the  first  portion  of  this  book  is  based  on  the 
principles  laid  down  in  that  paper.  The  solution  in  any  particular  case  becomes  easy 
if  the  end  fixing  moments  and  the  reactions  are  known,  and  values  of  these  have  been 
calculated  for  the  more  important  cases  likely  to  occur  in  practice. 

This  investigation  shows  that  the  values  of  the  various  moments  and  reactions  for 
a  given  loading  depend  on  the  relative  values  of  the  flexural  rigidity,  E  I,  and  the 
torsional  rigidity,  C  J,  of  the  section.  A  knowledge  of  the  geometrical  properties  of 
the  section  and  of  its  material  enable  the  former  of  these  to  be  predetermined  with 
some  accuracy,  but  the  authors  have  been  unable  to  find  any  published  data  as  to  the 
values  of  the  torsional  rigidity  for  such  commercial  sections  as  are  usual  in  structural 
engineering.  With  a  view  of  obtaining  such  data  experiments  have  been  carried  out 
by  Mr.  Ritchie  on  a  number  of  commercial  sections,  and  the  result  of  this  work  forms 
the  foundation  for  much  of  the  second  part  of  the  book. 

Chapter  I.  outlines  the  introductory  theorems  necessary  for  a  thorough  understanding 
of  Chapter  II.,  which  deals  with  the  equilibrium  of  the  bow-girder.  In  Chapter  III. 
the  torsion  of  non-circular  sections  is  considered,  while  Chapter  IV.  deals  with  the 
stresses  involved  by  such  torsion  alone  or  combined  with  bending,  and  Chapter  V.  deals 
briefly  with  the  general  principles  of  design  of  a  bow-girder  exposed  to  both  bending 
and  twisting. 

It  is  hoped  that  the  treatment  is  sufficiently  complete  to  enable  any  one  familiar 
with  the  general  principles  of  design  of  the  ordinary  straight  plate-web  or  lattice  girder 
to  adapt  these  to  any  specific  case  of  a  bow-girder  under  uniform  or  concentrated  loading. 

In  view  of  recent  failures  of  structures  in  which  straight  beams  exposed  to  some 
torsion  have  collapsed  under  seemingly  inadequate  loads,  the  data  of  Chapter  III., 
emphasising  as  it  does  the  extreme  weakness  of  the  commercial  I,  angle,  or  T  section 
under  torsion,  should  be  of  interest. 

Appendices  have  been  added,  giving  a  list  of  integrals  which  will  be  useful  to  the 
reader  working  through  the  investigations  of  Chapter  II.,  and  also  giving  a  table  of 
the  geometrical  properties  of  some  commercial  sections. 

A.  H.  G. 

E.  G.  R. 
DUNDEE, 

September,  1914. 

331443 


CONTENTS 


CHAPTER   I. 

AIMS.  I'A'.K 

1.  EQUILIBRIUM  OF  THE  STRAIGHT  GIRDER        ...                 ...  1 

2.  CURVATURE,  SLOPE,  AND  DEFLECTION 1 

3.  ENCASTRE  -AND  CONTINUOUS  BEAMS .         .  3 

4.  ENCASTRE  BEAM  WITHOUT  INTERMEDIATE  SUPPORTS       ......  4 

5.  ENCASTRE  BEAM  WITH  INTERMEDIATE  SUPPORTS 5 

(!.  ENCASTRE    BEAM    WITH    UNIFORM    LOADING — EFFECT    OF    SUBSIDENCE    OF  ONE 

SUPPORT     7 

7.  BEAMS  WITH  UNSYMMETRICAL  LOADING           ........  8 

8.  RESILIENCE  OF  A  GIRDER  UNDER  BENDING     ........  10 

0.  CASTIGLIANO'S  THEOREM          .                 .........  11 

JO.  RESILIENCE  UNDEK  TORSION  .         .         .         .         .         .         .         .         .         .         .12 

11.  DEFLECTION  DUE  TO  SHEAR  FORCES  13 


CHAPTER   II. 

12.  THE  CIRCULAR-ARC  BOW-GIRDER .  .14 

1P>.  CIRCULAR-ARC  CANTILEVER  WITH  SINGLE  END  LOAD 15 

14.  CIRCULAR-ARC  CANTILEVER  WITH   UNIFORM   LOADING  .         .        .        .        .        .16 

15.  CIRCULAR-ARC  ENCASTRE  GIRDER  WITH  SINGLE  LOAD 18 

16.  CIRCULAR-ARC  ENCASTRE  GIRDER  WITH  UNIFORM  LOADING 28 

17.  CIRCULAR-ARC  ENCASTRE  GIRDER  WITH  UNIFORMLY  LOADED  PLATFORM       .        .  34 

18.  GIRDER  WITH  UNSYMMETRICAL  LOADING        ........  37 

1!).  BOW-GIRDER  WITH  INTERMEDIATE  SUPPORTS 37 

20.  BOW-GIRDER  WITH  UNIFORM  LOADING  AND  CENTRAL  SUPPORT      .        .  37 

21.  BOW-GIRDER  WITH  TWO  INTERMEDIATE  SUPPORTS 40 

22.  BOW-GlRDER  WITH  THREE  INTERMEDIATE  SUPPORTS 43 

23.  EFFECT  OF  SUBSIDENCE  OF  SUPPORTS 45 

24.  EQUILIBRIUM  OF  A  COMPOUND  BOW-GIRDER 46 

25.  SHEAR  FORCE  AT  A  SECTION 47 

26.  EXPERIMENTAL  VERIFICATION  OF  FORMULAE 47 

27.  APPLICATION  TO  SECTIONS  OTHER  THAN  CIRCULAR 48 

CHAPTER   III. 

28.  THE  TORSION  OF  NON-CIRCULAR  SECTIONS 50 

29.  EXPERIMENTAL  RESULTS                                                   .  52 


\iii  COXTl'XTS 

CHAl'TKR    IV. 

new.  ''AUK 

:?(>.  SHKAI:  STKKSSKS  IN  A    BKA.M  OF  CIIMTKAK  SKITION       .  .         .       .r>0 

:'.l.  STUKSSKS  IN  XoN-(.1iucn.Ai:   SKCTIOXS     .....  .                  .       ">'.• 

:>±   HM.IPTICAL  SKCTIONS      ...  .       <'»0 

:;:;.   Ui;<TAN<;n.AU  AND  Box  SKCTIOXS  .         .  .       (II 

84.  I   SKCTLONS      .....  •       <;<s 

85.  IloitlXONTAL    SlIKAi:     IN    A     HKAM    I'NMKI:    ToliSIO.V     .  .          C.U 
:5C..    IvKSKKTANT    SlIF.AI!  .            .                        ...  .                                                           ('»!> 

37.  I  AND   Box    SKCTIONS     .....  .70 

CIIAl'THR    V. 

:•>«.    (JKNKUAK    I'lMNCIPKI'.S    OF     Dl'SKlN    OF    T1IK    BO\V    (Jlltl'i;!:  ...                          .          7o 


AiMMvXDIX    A. 
LIST  OF  INTKUKAKS       .............       77 

APPENDIX    I  '.. 
PROPERTIES  OF  CO.M.MKKCIAK  SKCTIONS.          ......  7s 

IXDKX  7!» 


A   STUDY   OF   THE    CIRCULAR- 
ARC   BOW-GIRDER 


CHAPTER   I 

(i)  Equilibrium  of  the  Straight   Girder. 

IF  a  girder  straight  in  plan  and  horizontal  when  unloaded  is  exposed  to  a  series  of 
vertical  loads,  each  section  is  subject  to  a  bending  moment  M,  whose  magnitude  varies 
from  point  to  point.  Under  the  influence  of  this  moment  the  girder  is  bent,  and,  so 
long  as  the  loads  are  not  sufficient  to  produce  stresses  in  excess  of  the  elastic  limit  of 
the  material,  the  radius  of  curvature  K  of  the  profile  of  the  neutral  axis  at  a  point 
where  the  bending  moment  equals  M  is  given  by  the  relationship 

1        M 
R  =  EI    .......     (a) 

where  I  is  the  moment  of  inertia  of  the  section  about  a  horizontal  axis  through  the 
centroid  of  its  area,  and  where  E  is  the  modulus  of  direct  elasticity  of  the  material. 
If  y  be  the  vertical  displacement  of  the  neutral  axis  at  a  point  distant  x  from  some 
datum  point  in  the  axis,  it  may  readily  be  shown  that 


=         (approx.) 

so  that,  so  long  as  the  deflection  of  the  beam  is  confined  within  practical  limits, 

d  M 


(2)  Curvature,  Slope,  and   Deflection. 

From  (lj)  it  follows  that  if,  at  any  one  point,  the  girder  is  horizontal  after  loading, 
the  slope  -r-  at  any  other  point  at  a  distance  I  will  be  given  by 

rl 

dil        |  M        .  t  . 

~  =      -777  .  dx  .         .         .         .         .     (c) 

dx  El 

Jo 

M 

and  will  therefore  be   represented  to  scale  by  the  area  of  the  yry  diagram  between 

Ji/J. 

the  two  points,  while  if  the  slope  at  the  first  point  is  not  zero,  this  area  -will  measure 
E.G.  B 


2  A   STUDY   OF   THE    CIRCULAR- ARC   BOW-GIKDKE 

the  difference  of  slope  at  the  two  points.  On  integrating  both  sides  of  expression  (<•), 
the  deflection  y  of  the  second  point  below  the  first  is  given  by 

fW ,  ,~ 

//  =        7-    dx (a) 

\dx/ 

Jo 

i  -i 
|.V        , 

I  /•;/  '  dx' 

*/ 0 

In  a  given  beam  under  load  the  slope  changes  from  point  to  point,  and  the  difference 
of  slope  at  two  points,  a  small  distance  Sx  apart,  is  given  by  -p  ( j-j  Sx,  or  by  f-^Jj 

M 

or  pj  .  Sx,  where  M  is  the  moment  acting  on  the  element  included  between  the  two 

sections.  If  the  rest  of  the  beam  were  to  remain  straight  the  deflection  at  a  distance 
I  from  the  element,  due  to  the  bending  of  the  element  under  this  moment,  would  be 
equal  to 

M       , 

-7TT    .     OX    .    I 

El 

and  if  the  slope  at  one  end  of  the  element  were  zero  this  would  be  the  actual  relative 
deflection  at  a  distance  I.  Since  every  section  of  the  beam  is  exposed  to  a  bending 
moment,  any  element  at  a  distance  x  from  the  point  whose  deflection  is  being  con- 
sidered contributes  its  quota  pj  .  x  .  Sx  to  the  resultant  deflection,  so  that  the  actual 

deflection  at  the  point  /,  relative  to  the  point  at  which  the  slope  is  zero,  is  given  by 

i 

M  j 

-TT7  .  x  .  dx 
EL 

o 

or  by  Ax    .......     (e) 

M 
where  A  is  the  area  of  the  -^j  diagram  included  between  the  two  points  and  x  is  the 

distance  of  its  centroid  from  the  point  /. 

If,  instead  of  being  zero,  the  slope  at  the  point  o  is  equal  to  i,  the  deflection  at  /, 
relative  to  this  point  is  given  by 

il  +  A  x          .  .         .     (/) 

Special  Cases  of  Deflection. 

In  certain  standard  cases  the  maximum  deflection  is  very  readily  calculated,  and  is 
as  follows  : — 

Deflection. 

Beam  of  uniform  section  and  of  length  /  with  single  load  W  at  centre        .       ajW  -777- 


r 
Beam  of  uniform  section  uniformly  loaded  with  u-  Ibs.  per  foot  run  .      ^§5  ... 

IJ'/:i 
Cantilever  of  uniform  section  with  load  W  at  end £   -rrr 

u-i4 
Cantilever  of  uniform  section  uniformty  loaded         .....       ^     - 


AND   CONTINUOUS   BEAMS 


(3)    Encastre"  and  Continuous  Beams. 

A  beam  simply  supported  at  its  two  ends  has,  everywhere,  a  curvature  whose  con- 
cavity is  upwards.  If,  however,  it  is  built  in  to  supports  at  its  ends,  these  supports 
prevent  the  beam  adopting  the  slope  natural  to  it  when  free,  and  a  fixing  moment  is 

W,  W0 


< 

w  k~~  & 

T 

&  -> 

"         3 

) 

r 

> 

f 

i 

t 

1" 

© 


© 


FIG.    1 . 


called  into  play  at  each  support,  these  moments  tending  to  make  the  beam  concave 
downwards.  The  effect  of  the  fixing  moment  is  transmitted  to  every  section  of  the 
beam,  and  at  any  such  section  as,  say,  X  in  the  beam  of  Fig.  IA,  for  equilibrium 

M,  =  Ma  -  (Eaxa  -  WM  -  W2x2)        .  .         .     (g) 

=  Mb  -  (Rbxb  -  Tf-V?,,) 


P. 


r— 


FIG.  2. 


P3 


while  at  X  in  Fig.  IB,  which  represents  a  beam  with  uniform  loading  of  magnitude 
w  Ibs.  per  foot  run, 


ii'  r 

a 


B    2 


4  A  STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 

If,  as  in  Fig.  2,  the  beam  has  one  or  more  intermediate  supports,  whose 
upward  reactions  are  PI,  P2,  PS,  the  moment  at  X,  say  between  supports  (1)  and  (2)  is 
given  by 

J/,  =  J/a—  l#a*a  + 


(j) 

-^f\        '  '  '       (k'} 

Under  such  circumstances,  the  magnitudes  of  the  fixing  moments  Mn  and  Mb ;  of  the 
reactions  Ra,  Rb,  at  the  ends  ;  and  of  PI,  P2,  PS,  the  reactions  at  the  supports,  require 
to  be  determined  before  equations  (g),(h),(j),  or  (k)  can  be  used  to  determine  the  value 
of  MS  at  any  given  section. 

(4)  Encastr£  Beam  with  no  Intermediate  Supports. 

Considering,  for  example,  the  case  of  a  beam  built  in  at  its  two  ends  and  carrying 
a  uniformly  distributed  load  of  w  Ibs.  per  foot  run  (Fig.  3), 

2 

Mx  =  Ma  -  Rax  +  -r-. 


U 


1— 

FIG.  3. 


Since  from  symmetry  Ra  =  ->  we  nave 


WLX 


If,  for  simplicity,  the  section  of  the  beam  be  taken  as  constant  so  that  Ix  =  constant 
=  I,  we  have,  on  integrating, 

(]l-  JLIjif  x. 

dX  JltJ.      V 

where  A  is  a  constant  of  integration.  Since  the  slope  is  zero  where  x  =  0,  i.e.,  at  the 
left-hand  support,  it  follows  that  A  =  0,  and  since  the  slope  is  also  zero  at  B  where 
x  =  1,  we  have 


ivtw  _j_  A 

4    ^ 


Substituting  this  value  of  A/a  in  (m)  gives 

dx 


1C 

Tl  112 


ENCASTRti   BEAM   WITH   INTERMEDIATE   SUPPORTS        5 

or,  on  integrating  this, 

jv_  {Px       x^  _  W          } 
"  El  112  +  24       12"*      I' 

The  constant  B  is  determined  from  the  fact  that  the  deflection  y  =  0  when  x  =  0,  so 
that  .6  =  0. 

(5)  Encastre"  Beam  with  Intermediate  Supports,  or   Continuous   Beam    on    more 

than  Two  Supports. 

Let  A,  B,  C,  (Fig.  4)  represent  three  adjacent  points  of  support  on  an  encastre 
beam,  or  on  a  simple  continuous  beam  with  uniform  loading  iv  Ibs.  per  foot  run.  To 
determine  the  moments  Ma,  Mb,  and  Mc,  and  the  reactions  Ra,  Rb,  Rc.  Take  the 
origin  at  A.  Then  between  A  and  B, 

Mx  =  Ma  —  Rax-\-~ (n) 


.'.  At  B 


M  =  M  - 


r/i2 
2 


.     (o) 


B 


FIG.  4. 


Similarly,  working  back  from  C  to  B, 


Writing  (>?)  as 
we  get 
and 


dy  _ 


war 


•         •     (P) 


a  O  2i^± 

Since  //  =•  0  when  a:  =  0,  it  follows  that  D  =  0,  while  since  y  =  0  when  x  =  1, 
we  have 

Tl  f-     ^1_ -D      ^1 I       ^'^1          I       /^»7      Q 

•"*«    Q  n     (*        I        41/1      ~T~    v*l  v 


''•  C  ~        '2  6     +  24  [ 


From  (5)  the  slope  at  B  is  given  by 


and  on  substituting  from  (s)  this  becomes 

fdy\          1    f      /i 
\d~xj  ~m  I     "2 


. 

o     1  • 

8  J 


6  A   STUDY   OF   THE   CIECULAE-AEC   BOW-GIBDER 

Similarly,  taking  C  as  origin  and  working  back  from  C  to  B,  we  should  get 


b     hi  (     t.2  3         8 

the  minus  sign  being  taken  before  y-_,  because  x  is  now  measured  in  the  negative 
direction. 

Equating  these  expressions  for  (-—-  J  ,  and  eliminating  terms  containing  R<t  and  7?(. 

\  M  •  /  /    fa 

by  substitution  from  equations  (p)  and  (o),  we  get 


the  relationship  commonly  known  as  the  equation  of  "three  moments."  With;? 
points  of  support  this  theorem  yields  n  —  2  equations,  and  the  terminal  conditions 
supply  the  additional  two  which  are  necessary  before  the  n  unknowns  can  be 
determined. 

Taking,  for  example,  the  case  of  a  beam  resting  on  three  equidistant  supports  and 
forming  two  spans  each  of  length  /,  J/,,  —  ~MC  =  0,  and  the  foregoing  equation  reduces  to 

Mb  =  -g-. 

Also  since  Mb  =  Ma  —  I\J  + 

-i 


.'.  Rb  =  2  id  -2  !!„=,  id. 
8 

Again,  taking  the  case  of  an  encastre  beam  with  a  central  support  giving  two  spans, 
each  of  length  /,  from  symmetry  Ma  =  Mc,  ^=il2  =  I,  and  equation  (u)  becomes 


From  (o)  Mb  =  3/rt  -  Ral  +  " 

.'.  3  M  -  2  Ral  +  |  id2  =  0 

Since  the  slope  at  A  where  x  =  0  is  zero  it  follows  from  (q)  and  (s)  that 

Mal      RJ*  ,   id3  _ 
~2~       ~6"   h24~ 

.-.  83/a  -  RJ,  +  ^  =  0 

and  combining  equations  (»•)  and  (x), 

/,   ._  «'1 
lL"-  2 

/.  Rb  —  2  u-l  —  2  7^,  =  «•/. 

/r/2       ?r/2       »-/2 
From  :;.!/„  =:  -  ---        :  -- 


from  (v) 


ENCASTKti   BEAM   WITH   UNIFORM   LOADING 

id2      Mn 


1*2 ' 


(6)  Encastre"  Beam  with  Uniform  Loading  —  Effect  of  a  Settlement 
of  One  Support. 

Where  the  fixing  moments  Mn  and  Mh  at  the  ends  of  an  encastre  beam  of  span  I, 
or  at  any  two  intermediate  supports  of  a  continuous  beam,  are  not  equal,  the  moment 
due  to  these  varies  uniformly  from  Ma  to  3/,,,  and,  at  a  point  distant  x  from  the  end 
A  ,  is  equal  to 


From  equations  (</)  and  (A)  (p.  3),  it  is  evident  that  in  a  loaded  beam,  fixed  at  the 


FIG.  5. 

ends,  the  bending  moment  at  any  point  is  the  difference  between  the  bending  moment 
which  would  be  produced  by  the  same  loading  on  a  beam  simply  supported  at  the  ends, 
and  that  produced  by  the  end  moments,  so  that  in  the  case  of  a  uniformly  loaded 
encastre  beam  with  end  moments  Ma  and  Mb  the  diagram  of  effective  bending  moments 
is  represented  by  the  shaded  area  of  Fig.  5. 

If  one  of  the  supports  A  of  such  a  beam  sinks  through  a  distance  d,  the  ends 
remaining  horizontal,  the  difference  of  slope  of  the  ends  is  zero,  and  consequently 
from  (c)  (p.  1),  the  area  of  the  effective  B.M.  diagram  is  zero. 


2 
/.  Ma 


"S    '  8 
Mb  =  f . 


Again,  since  the  relative  deflection  of  the  two  ends  is  d,  the  moment  of  the  effective 
B.M.  diagram  about  A  is  equal  to  Eld  ((e)  (p.  2)). 


A   STUDY   OE   THE   CIBCULAR-AKC   BOW-GIRDER 

'  2  l  l  U'1*          2  l 


=  (Mn  +  2  37,,)      - 

GEId.wP 

.  .  M,,  +  2  Mb  =  —  p  --  1-  — 

7/;/2 

and  Ma  +  Af6  =  ~ 

,,  6Eld      wP 

/2       h  12 


_  . 

~F"     "12 

If  the  only  moment  is  that  induced  by  the  settlement  of  the  support,  i.e.,  ic  =  0, 

I.T--    r>  T>*  6EId 

this  5.M.  =  +        2    . 

The  reactions  Ra  and  L'6  under  the  new  conditions  are  determined  from  the  equa- 
tions — 

w/2 
Mb  =  M,,  -  linl  +  ~ 

M   -  Mb   .   wl 

~  ~z 

_w±       12EH 
'   2  Is 

and  /4  =  «•/  —  A',, 

TT         IZEId 


(7)  Beams  with  Unsymmetrical  Loading,  or  with  a  Series  of  Concentrated  Loads. 

When  the  loading  of  an  encastre  or  continuous  beam  is  unsymmetrical  or  consists 
of  a  series  of  concentrated  loads,  a  semi-graphical  treatment  based  on  the  considera- 
tions outlined  on  p.  2  is  preferable. 

In  Fig.  6,  let  A,  B,  C  be  three  adjacent  supports  in  a  continuous  beam,  and  let 
AGB,  BHC  represent  the  bending  moment  diagrams  for  such  a  loading  on  two 
simply  supported  spans  AB  and  BC.  Let  ADEB  and  B EFC  represent  the  fixing 
moment  diagrams,  and  G^,  G\,  G2',  G2  the  positions  of  the  centroids  of  the  areas 
A  I>EB,  AGB,  BE  EC,  BU( '. 

Let  the  area  AGB  =  AI 
„     ADEB  =  A,' 
,,         „        BHC  =  A  2 
„      BEEC=  A2' 

Then,  considering  the  span  AB,  taking  .4  as  origin,  since  the  supports  at  .4  and 
7)  are  at  the  same  level 


ih  being  the  slope  at  B. 


BEAMS   WITH  UXSYMMETKICAL   LOADING 


9 


Similarly  for  the  span  CB,  taking  C  as  origin,  since  the  supports  at  C  and  B  are 
at  the  same  level 

1 


o 

r 


0 


K-   a"     — A 


the  negative  sign  being  taken,  since  x  is  measured  in  opposite  directions  in  the  two 

cases. 


10          A   STUDY   OF   THE   CIKCTJLAR-ABC   BOW-GIEDEE 

Equating  the  two  expressions  for  the  slope  at  13  gives 

f±i^i__zuli£i  _  _    2  ~*2  ~  ^z''-i  /{\ 

fcj  /2 

Again,  taking  moments  about  .4    and   C  of  the  fixing  moment  diagrams  on  each 
span 


and,  on  substituting  these  values,  equation  (/;)  becomes 

M,J,  +  237,  (^  +  /2)  +  3/,./2  -  6  {^i  +  ^|  -  0      .         .         (*) 

'i  '2 

This  is  the  most  general  form  of  the  equation  of  these  moments  and  is  applicable 
to  any  form  of  loading, 

9    trl  3  9    irJ  3  /  / 

1TT     •!'  a     UJt-i  .  £     II  in  (i  ' 

Writing  Ai  =  -  -gL  ;  ,18  =  --£-;  x,  =  ^  ;  a-2  -  ^ 

gives  the  equation  for  uniform  loading,  which  is  identical  with  (u),  p.  (5. 

If  some  or  all  of  the  supports  sink,  B  falling  d^  below  A  and  d.2  below  C,  equation 
(z)  becomes 

MJ,  +  2.V,  (I,  +  g  +  M,.l,  -  C>  j4jfi  +  ±^2  1 

'l  '2 

(a) 

(8)   Resilience  of  a  Girder  Exposed  to  Bending. 

If,  under  the  action  of  a  bending  moment  M,  two  originally  parallel  vertical  sections 
of  a  beam,  enclosing  an  element  of  length  bx,  become  inclined  to  each  other  at  an  angle 

rj  • 

of  Si,  the  work  done  by  the  moment  in  bending  this  element  is  equal  to  M  -  '.     (This 

2i 

assumes  that  the  moment  is  applied  gradually,  and,  at  any  instant,  is  proportional  to 
the  curvature  obtaining  at  that  instant.) 

.'.  "Whole  work  done  in  bending  beam=  ^  ---.     M<ii,  where  the  integration  is 

taken  over  the  whole  length  of  the  beam. 

But  Si  measures  the  difference  of  slope  at  the  two  ends  of  the  element. 

Bx       M 

•'•K=~R=~E2  ** 

So  that  if  /  be  the  length  of  the  beam 

t    /V,, 

09) 


This  quantity  is  termed  the  resilience  of  the  beam  under  the  given  loading,  and  is 
equal  to  the  work  done  by  the  load  or  loads  during  the  distortion  of  the  beam.  Thus, 
if  a  single  load  W  be  applied  to  the  beam,  causing  its  point  of  application  to  deflect 

y 

through  a  distance  y,  the  work  done  by  it  during  its  application  is  equal  to  JC^. 


CASTIGLIAXO'S   THEOEEM 


11 


E.;/.,  beam  simply  supported  at  the  two  ends.     Single  load  W  at  a  point  C  distant 
a  from  one  end  and  b  from  the  other  end  of  the  beam  (Fig.  7). 

Rb  =  W 
Wbx 


Here  II a  =  W 
Between  A  and  C,  J\IX  = 
1 


2A7      Va  +  b 


dx 


FIG.  7. 
Similarly  between  B  and  C 

^(6-c)  =  6h7(aH 


'3EL  (a  +  b) 

(9)  Castigliano's  Theorem. 

Where  more  than  a  single  load  is  applied,  the  problem  is  readily  solved  by  an 
application  of  this  theorem. 

Suppose  a  structure,  originally  horizontal,  to  deflect  through  yl  and  yz  at  points 
Px  and  P2  under  the  application  of  loads  Wl  and  IV  z  (Fig.  8).  Then  assuming  smooth 
supports,  so  that  the  work  done  by  the  end  reactions  is  zero,  we  have  — 


Tl   - 

u  — 


i 


Let  TFj  be  now  increased  to  (]V1  +  8}\\),  W2  remaining  constant,  and  let  y±  + 
2/2  H~  ^/2»  ^e  ^ne  new  deflections. 


12         A   STUDY   OF   THE   CIECtJLAB-AEC    130W-GIEDEE 


The  additional  work  done  = 


(  £>  TIr  ^1 

\  W±  +  -  ^  \  S//x  +  Tr28//2 

{.  &   } 


.-.&U=  ir^'/i  +  W£yt  .     (y) 

Now  suppose  the  structure  unstrained  and  gradually  loaded  with  (W\  -f  ^^\) 
and  TT'2,  these  loads  during  application  always  maintaining  towards  each  other  the 
ratio  of  their  final  values.  The  final  deflection  must  be  the  same  as  before,  while  the 
resilience  is  given  by 


U'  = 


i  +  Tr2?/2)  +  y 
=  i  {2^+517  +  ^8^ 

But  U'  —  U  must  equal  &U. 

.-.  8[7  =  ?/ 

dU 


TF 


.     (7) 


a-     -i     i     (H 
Similarly  -^  =y2, 


derivative  of  U  with  respect  to  any  one  load  equals  the 


deflection  of  the  point  of  application  of  that  load. 

(10)  Resilience  of  a  Beam  Exposed  to  a  Torque. 

If  a  beam  be  exposed  to  a  torque  whose  magnitude  at  a  given  point  is  T,  successive 
plane  sections  suffer  rotation  about  the  longitudinal  axis  of  the  beam,  and  the  relative 
rotation  of  two  sections,  distant  Bx  apart,  is  equal  to  86,  where 


Here  ('  is  the  modulus  of  transverse  rigidity  or  the  shear  modulus  of  the  material 
and  J  is  the  polar  moment  of  inertia  of  the  section,  or  its  moment  of  inertia  about  an 
axis  through  its  centroid  perpendicular  to  its  plane.1 

5j/3  Y'2 

The  work  done  by  the  torque  during  this  relative  rotation  is  T  -         ^—  bx,  so  that 

Z         ZCf7 

over  the  whole  length  I  of  the  beam  the  work  done  by  the  torque  is  given  by 

? 
7-2 

<  j  ''' 


1  SeelChapter  III.  fur  the  effective  value  of  J  in  any  particular  case. 


DEFLECTION   PRODUCED   BY   SHEAR  FORCES  13 

Where  a  beam  is  exposed  to  both  bending  arid  twisting  moments,  its  resilience  is 
the  sum  of  the  works  done  by  these  moments,  and  this,  by  the  principle  of  work,  is 
equal  to  the  work  done  by  the  applied  loads  daring  distortion. 

(u)  Deflection  Produced   by  Shear  Forces. 

In  addition  to  the  deflections  produced  by  the  bending  of  a  girder,  there  is  some 
slight  deflection  due  to  the  fact  that  each  vertical  layer  is  exposed  to  shear  stress.  In 
a  straight  beam,  exposed  only  to  bending  and  shear  stresses,  the  deflection  due  to  shear 
is  always  a  small  fraction  of  that  due  to  pure  bending,  being  greatest  in  a  built  up 
beam  of  I  section  in  which  the  web  is  comparatively  thin.2 

In  such  a  beam  of  normal  proportions  and  span  simply  supported  at  the  ends,  the 
deflection  due  to  shear  is  seldom  more  than  4  or  5  per  cent,  of  that  due  to  pure 
bending.  In  an  encastre  beam  of  this  type  the  proportion  may  be  as  much  as  20 
or  25  per  cent.  In  the  type  of  bow-girder  to  which  this  treatise  is  particularly 
devoted,  the  deflection  is  mainly  due  to  torsion,  and  moreover  the  proportion  of  the 
whole  deflection  due  to  torsion  is  greatest  for  those  beam  sections  for  which  the  shear 
deflection  is  greatest.  Even  in  an  extreme  case,  in  a  bow-girder  the  shear  deflection 
does  not  amount  to  more  than  4  or  5  per  cent,  of  the  whole,  and  will,  in  general,  be 
much  less  than  this.  It  has,  in  consequence,  been  neglected  in  the  following  dis- 
cussion. Where,  as  in  a  large  built-up  bow-girder  of  I  section  with  very  slight 
curvature,  it  may  be  advisable  to  make  allowance  for  the  extra  deflection,  this  may 
most  easily  and  with  sufficient  accuracy  be  taken  into  account  by  using  in  the  calcula- 
tions a  value  of  E  about  20  per  cent,  less  than  the  true  value  for  the  material . 

2  For  a  discussion  of  this  point,  see  Morley's  "  Strength  of  Material?,"  p.  226,  or  any  text-book 
on  the  same  subject. 


CHAPTER   II 
(12)  The  Circular-Arc  Bow-Girder 

A  GIRDEK  built  in  to  supports  at  one  or  both  ends  and  forming  an  arc  of  a  circle 
in  plan,  is  subject,  at  each  section,  to  both  bending  and  twisting  moments.  At  the 
supports,  fixing  moments  of  both  kinds  are  called  into  play,  and  until  these  are  known 
the  resultant  moment  tending  to  cause  rupture  at  any  section  is  indeterminate.  The 
following  investigation  is  devoted  to  a  consideration  of  the  general  state  of  elastic 
equilibrium  of  such  a  girder  under  various  systems  of  loading. 

The  investigation  is  based  on  the  theorem  (p.  1)  that  in  a  straight  beam,  fixed 
horizontally  at  some  point,  the  slope  at  any  other  point  is  given  by  the  area  of  the 

M 

rjj  diagram  between  the  two  points.  Where  a  girder  is  circular  in  plan  and  is  sub- 
jected to  both  bending  and  twisting  moments  this  theorem  requires  modification. 
Let  M0  and  Te  be  the  bending  and  twisting  moments  at  a  point  P  distant  6  (in  angular 
measure)  from  the  support  A  (Fig.  9).  Then  a  given  slope  at  P  in  the  direction  of  the 
tangent  at  this  point  produces  a  slope  of  cos  (6^  —  6)  times  its  magnitude  at  Q  in  the 
direction  of  the  tangent  at  Q.  Also  an  angular  displacement  7  at  P,  due  to  a  torque 
between  the  support  and  this  point,  produces  a  slope  y  sin  (61—  9)  at  Q,  in  the  direction 
of  the  tangent  at  Q. 

It  follows  that  if  distances  along  the  arc  of  the  girder  be  represented  by  s,  the 
resultant  slope  at  Q,  assuming  the  slope  at  the  support  to  be  zero,  is  given  by 


/»arc  61 

(dy\    =  I  jM* 
\ds/0i       ]  EI0 

•f    A 


arc  61  /*arc  61 

Mfi  cos  (0!  -  0)  ds  +    — ^  sin  (0i  -  e) ds 


Here  I6  and  J6  are  the  moments  of  inertia  of  the  section  at  0,  about  the  axes  of 
bending  and  of  twisting. 

Where  the  beam  is  of  uniform  section,  this  becomes 

(^)    =  _L       Me  cos  (0!  -  0)  ds  +  ~     Te  sin  (0!  -  0)  ds ; 
\ds/8i       El  I  CJ  I 

v  */  * 


or,  sinceKif  r  is  the  radius  of  the  arc, 


.     dy       1     dy 
ds  =  rd6  ;  -=-=-.  -y^ ; 
'   ds       r     eld 


CIECULAE-AEC  CANTILEVER  WITH  LOAD  W  AT  FEEE  END        1  5 


(13)  Circular-Arc  Cantilever  with  Load  W  at  Free  End. 

Let  a  (Fig.  9)  be  the  angle  subtended  by  the  arc. 

Then, 

Me  =  W  X  CR  =  Wr  sin  (a  —  0), 

Tg  =  W  X  RP  =  Wr{l  —  cos  (a  —  0)}  ; 


...  W)     =  !IT  |  sin  (a  -  8}  cos  (0,  -  0)  </0  +  ~  I  {1  -  cos  (a  -  0)}  sin  (0!  -  0)  (10 

On  integrating l  and  simplifying,  this  becomes 

id-u\         in3  T  1 

~T7.      =  — rr?     0i  sin  (a  —  0i)  +  sin  0i  sin  a     + 
W0M       2£I  L  J 

^r  |2  (1  —  cos  0i)  +  0i  sin  (a  —  00  —  sin  0i  sin  a  .         .     (1) 

^  V,  t/     l^  — J 


FIG.  9. 


As  61  is  any  angle  between  o  and  a,  on  writing  0X  =  d  in  this  expression  and  integrating 
between  the  limits  61  and  o,  we  get  the  deflection  at  dv 

fOl 

Wr3  I 
/.  7/(e])  =:——     {0  sin  (a  —  0)  +  sin  0  sin  a}  c?0  + 

^o 
/•fli 


{2  (1  —  cos  0)  +  0  sin  (a  —  0)  —  sin  0  sin  a}d0 
T'JT?*3  r~ 

=  ^ry-ry        01  COS  (tt  —  0l)    —  COS  «  Sin  01 

2ilLl  L 

.)  +  0!  cos  (a  -  0i)  + 

sin  (a  —  0i)  +  sin  a  (cos  0i 


r8  T2  (0!  -  sin  6^  +  0i  cos  (a  -  6$  +  "I 

V  L  sin  (a  —  #1)  +  sin  a  (cos  0i  —  2)  J 


(2) 


At  the  free  end  d±  =  a,  and  wre  have 

Wr3  r  "I        Wr3  P  ~l 

yw  =  njT-T     a  —  cos  a  sin  a     -\-  ~— —     8a  —  4  sin  a  +  sm  a  cos  a  .     (o) 

2iJijL  L  J        2C't/  L  -J 

As  a  check  on  the  validity  of  the  reasoning  leading  to  these  results  the  deflection 

1  For  convenience  in  integrating  this  aad  other  expressions  occurring  in  the  course  of  this 
investigation,  a  list  of  the  necessary  integrals  is  given  in  Appendix  A. 


16 


A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


at  the  weight  may  be  calculated  by  equating  the  resilience  of  the  beam  to  the  work 
done  during  deflection.     Taking,  for  convenience,  the  origin  at  the  free  end  (Fig.  10), 

M  =  Wr  sin  0  ;  Te  =  Wr  (1  —  cos  6)  ; 
and,  if  I  be  the  length  of  the  beam,  the  resilience  is  given  by  — 


2EI 


Tjds  = 


-  2  cos 


_  W2rs  [~a  —  cos  a  sin  a       3a  —  4  sin  a  -f  sin  a  cos  a"| 
~T~  El  CJ  J  ' 


\v 


and,  since  this  =  -  -  X  deflection  at  weight, 

' 


FIG.  10. 


Wr3  fa  —  cos  a  sin  a  3a  —  4  sin  a  +  sin  a  cosal 

•"•  lJw  =  -5-                     7/r  ni 

5a     L  •           X"  ^"                         —^ 

which  is  identical  with  equation  (3). 


a  =  ~  =  90°, 

a 


uv  r  -n      i-sir  -  q  __     3  r-7854    56621 
•7^  =  T  L2E7  +  ~CAT^         ]}1  LET       CJ  J  • 


a  =  ~  =  135°, 
4 


ir/- 


16 


377   +   2      .      977  _  -^  _  2 

v  2 


4CJ 


=  nv 


1-4281       1/8716 
AT  CV 


(14)  Circular- Arc  Cantilever  with  Uniform  Loading— w  Lbs.  per  Unit  Length 

Taking  the  origin  at  the  free  end,  we  have,  at  any  point  0  (Fig.  11) 

re 
Me  =     wrz  sin  <j>d<f>  =  «T2(1  —  cos  V), 

Jo 

CO 

Te  =     wr*(l  —  cos  </>)  d<f>  =  wi*  (0  —  sin  0), 
Jo 


CIECULAE-AEC  CANTILEVEE  WITH  LOAD  W  AT  FEEE  END      17 

as  the  moments  produced  at  P  by  the  loading  on  that  portion  of  the  beam  between 
P  and  the  free  end. 


h/\      _  wr 


wr 


where  a  is  the  total  angle  subtended  by  the  beam.     Integrating  this  expression  and 
simplifying  gives 


A  „  fa  —  0i  .  sin2a—  sin20i)      sin  0i  sin2  a—  sin30i 

m  (a  _0;)_COs  - 


sin  (a  —  0i)  —  a  cos  (a  —  0i)+  0i  —  cos0i 


a—  0i      sin  2a—  sin20 


] 

''} 


sin  0i  sin2  a  —  sin3 


fa— 0      sin  2a— sin  20)      sin0sin2a  —  sin30"1  ^ 


sin(a  —  0)  —  acos(a  — 


fa  —  0      sin2a—  sin20T 
"~ 


sin  6  sin2  a  —  sin3  0 


2AT 


2  —  2  cos  (a 


--. 

26'e/ 


-  |  (cos3  a 

2  —  2  cos  0 


.  sin  &i  sin  2a 
>i)  +  (a  —  0i)  sin  0i  +  cos  a  —  cos  0i  +  -      — 5 — 

a  —  0i       sin  2a  —  sin  20i 
a  —  cos3  0i)  —  cos  0i  snr  a  -\ ^ —  . 


cos  0i  sin2  a 

0i)  +  a2-  012  +  (a-0i)sin0i 

sin  0i  sin  2a   .    ,  ,      a 
+  cos  a  —  cos  0i -^—    -  +  |  (cosd 

a  —  0,    .   sin  2a  —  sin 
+  cos  0j  siir  a  --     — ^  — |— 


2  cos  (a  —  0i)  —  2a  sin  (a  —  6 

„        sm0lSin-za   ,    -     ;3  a  _  COS3  ^ 


B.G. 


18  A   STUDY   OF   THE   CIECULAU-AHC   BOW-GIHDEK 

At  the  free  end  0t  =  0,  and  the  deflection  becomes 

ICIA   [V                                                                      a    .    sin  2a~| 
typj     1  —  cos  a  —  f  (cosja  —  1)  —  sin- a  +  -  -\ 7— 

.    tnA    P1                                                                              -    .,         a      sin  •_>«"] 
-)-         •  I  1  —  cos  a — 2a  sin  a  -j-a  +  5 (COS8a  —  1)  +  sin  a  —  -  H — 

1_(   •/    L  !u  4       J 


e.g.,  if  a  =  5,  the  deflection  at  the  free  end  becomes 

a 

4  f-5594   .   -1035 

"»  =  "•'   .^r  +  ^r 


(15)  Circular- Arc  Girder,  Built  in  at  Two  Ends,  with  Single  Load   W. 

Let  the  arc  subtend  an  angle  (n  —  2  <£),  and  let  0  (Fig.  12)  be  its  centre  ;    Alt 
the  line  of  supports;  AOW  =  a;  BOWT=  @;  It,,  and  1\,,  the  vertical  reactions  at  A 


Fio.  12. 

and  1?  /  3/ft  and  Jl//,,  T,,  and  7',,  the  bending  and  twisting  moments  at  the  supports  A 
and  B,  the  axes  of  these  moments  being  respectively  parallel  to  and  perpendicular  to 
OA  and  OB. 

The  bending  and  twisting  moments  at  any  point  between  A  and  W,  distant  6 
from  OA,  are  now  given  by 

Me  =  ^fll  cos  e  —  ll,,r  sin  B  +  7',,  sin  6 (4) 

TS  =  Tn  cos  0  +  Iiar  (1  —  cos  0)  —  3/fl  sin  0      .         .         .     (5) 

while  the  moments  at  a  point  between  B  and  W,  distant  6  from  OB,  are  given 
by  similar  expressions,  with  suffix  b  taking  the  place  of  suffix  a. 

Before  these  moments  can  be  calculated  for  any  particular  case,  the  values  of 
the  six  unknowns,  M(l,  M,,,  7',,,  T,,,  It,,,  11,,,  are  to  be  ascertained;  and  for  this,  six 
relationships  between  these  unknowns  are  necessary. 

Taking  moments  about  B,  of  the  forces  and  couples  acting  in  a  vertical  plane  we 
have,  for  equilibrium, 

E,,  ('2r  cos  <£)  —  Ta  cos  </>  —  M/r  sin  cjb  —  IF;-  •  cos  </>  +  cos  (a  +  0)1  -f  'J'i, cos  </>  -f  ^  sin  ^  =  0 


CIRCULAR-ABC   GIRDER,    BUILT   IX   AT   TWO   ENDS        19 


•  en 


Again,  taking  moments  about  the  line  AB, 

(Ma  +  Mb)  cos  <£  -  (77a  +  Tb)  sin  <£  =  Wr  {  sin  (a  +  </>)-  sin  </>}.         .     (8) 

while,  equating  the  torques  at  the  weight,  as  obtained  by  working  from  both  ends  of 
the  girder, 

Tn  cos  a  +  Rar  (1  —  cos  a)  —  Ma  sin  a  =  —  Tb  cos  ft  —  Rbr  (1  —  cos  ft)  + 

Mb  sin  /3          ............     (9) 

The  other  two  necessary  relationships  are  obtained  by  expressing  the  fact  that  both 
slope  and  deflection  at  the  weight  are  the  same,  whether  the  latter  is  considered  as 
being  at  one  extremity  of  the  arc  A  W,  or  of  the  arc  BW. 
The  slope  at  any  point  0X  between  A  and  W  is  given  by 


cos    - 


sn    - 


and,  on  substituting  for  Jl/fl  and  Te  from  (4)  and  (5)  and  integrating, 

sin  6,\  -  (Rar  -  Ta]  0,  sin  ^ 


cos 


Similarly  at  any  point  between  B  and  IF,  distant  0X  from  OB, 
"  cos  0X  +  sin  0j  I  —  (Ebr  —  Tb)  6l  sin  0t 

—  /4r)  01sin  0X  +  2/4r(l  —  cos  0X)  —  M6  -[  sin  0X  —  0X 

The  slope  at  the  weight  is  obtained  by  writing  6l  =  a  in  the  first,  or  0X  =  ft  in  the 
.second  of  these  expressions,  and  is  thus  given  by 


/•2   r  n 

.,-777  I  M,<  i  a  cos  a  +  sin  a  }  —  (Rar  —  Ta)  a  sin  a 


r2  r 

^7-7     (7Tft 


cosa) 


sna  —  acosa 


n 

I 


(10) 


or  by 


cos 


sn      ~      T  ~ 


sn 


r2    f  "1 

+  0777    (T6  — 74r)/3sin/3  +  2/4r(l—  cos/3)—  A/6{sin/3  — /3cos/?l 

ZCe/    l_  '  J 

according  as  the  point  IF  is  considered  as  forming  part  of  span  AW  or  of  span  B  W. 
On  equating  these  two  expressions,  with  the  sign  of  the  second  changed  since  0 

c  2 


20 


A   STUDY   OF   THE   CIECULAE-AEC   BOW-GIEDEE 


is  measured  in  opposite  directions  in  the  two  sections,  a  further  relationship  between 
the  unknowns  is  obtained. 


Deflections.  —  Assuming  the  supports  to  be  at  the  same  level,  integrating 
obtain  the  deflection  gives  (between  A  and  W) 


to 


r^_  C°r 

IK  I]    L    a 

Jo 


-  (Rar-  ra)0sin0J  dd   • 

ar(l  —  cos  0) — 3/ft !  sin  0  —  0cos  0  M 

IV  /  (I   (  (  J 


1 .  2G'J 

as  the  deflection    at    a   point  distant  61  from  A.     On  integrating  and  simplifying, 
this  becomes 


[  A^  sin  0X  -  (Rar  -  TJ  (sin  0,  -  d,  cos 


7Tf(  _  jRar)(sin  0X  —  01cos  0^  +  27^'  (^i  —  sm  ^i 
+  Ma(018in01+2eo801— 2) 

Similarly  for  a  point  between  B  and  W,  distant  6l  from  B, 


J  sin  0X  -  (/V  -  Tfc)  (sin  0X  -  01  cos  0 


(12) 


2EI 


(Tb  -  /V)  (sin  0t  -  #1  cos  ^)  +  27^(0!  —  sin  6 


r2     r( 

h  2CJ  L  +  3/6  («i  sin  0!  +  2  cos  ^  -  2) 


(18) 


At  the  weight,  0X  becomes  a  in  (12)  and  /3  in  (13)  and  these   expressions  give 
(A  to  W) 


r2     f  "1 

JJTJ-    Maa>  sin  a  —  (Ear  —  Ta)  (sin  a  —  a  cos  a)  J 

r2     r(!rrt  —  Ear)  (sin  a  —  a  cos  a)  +  2/O'  (a  —  sin  a) 
2CV  L  +  ^/«  (a  s"1  a  +  2  cos  a  —  2) 


•  (14) 


and  (B  to  IF) 


sD 


2A7 


Mb  /S  sin  /3  -  (726r  -  T6)  (sin  /3-j3cos 
r(Tb  -  J?br)  (sin  /3  -  /3  cos  /3)  +  2 


sn 


1 


(15) 


On,  equating  the  identities  (14)  and  (15)  the  final  relationship  is  obtained,  and 
from  the  six  equations  (6),  (7),  (8),  (9),  (10  =  --  11),  (14  =  15),  the  six  unknown 
fixing  moments  and  reactions  may  be  determined  in  any  particular  case.  These 
moments  depend  somewhat  on  the  relative  values  of  El  and  of  CJ,  except  where  the 
load  is  in  the  middle  of  the  span.  An  increase  in  the  ratio  El :  CJ  is  accompanied  by 
an  increase  in  all  the  fixing  moments.  The  effect  on  the  values  of  Ma,  of  Mb,  and  of  the 
end  reactions,  produced  by  a  large  variation  in  this  ratio,  is  very  small,  especially 
when  the  angle  a  is  large.  The  effect  on  the  end  torques  is  more  pronounced, 
particularly  for  small  values  of  a. 

In  order  to  facilitate  the  application  of  the  results  of  this  analysis,  and  to  make 


CIKCULAB-AKC   GIEDEE,   BUILT   IN  AT   TWO   ENDS         21 


10 


1/sL/ues  of  (i 

fi-MA 

FIG.  13.— Values  of  MA,  MB  and  EA  for  a  bow  girder  built  in  at  both  ends,  subtending  an  angle 
180°  —  2<f>,  and  carrying  a  load  If  at  a  point  distant  a  from  end  A. 


A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


30°  40'  50" 

\fa.lues  of  K 


60 


70 


80 


FlG.  14. — Values  of  J/A,  J/H  and  A'A  for  a  bow  "inlor  built  in  at  both  ends,  subtending  an  arc 
(180°  —  2<f>},  and  with  a  single  weight  II'  distant  a  from  the  end  A. 


CIECULAR-AEC   GIBDEB,   13UILT   IX   AT   TWO   ENDS 


23 


it  more  useful  in  practice,  the  foregoing  equations  have  been  solved  for  a  series  of 
values  of  a  and  of  <£,  and  the  values  of  the  end  moments  and  reactions  have  been 
calculated  for  a  series  of  values  of  El:  CJ.  Owing  to  the  comparatively  small 
effect  of  this  ratio  on  the  end  bending  moments  and  reactions,  values  of  these  have 


40'  50' 

Values  of  (L 

FIG.  15. — Values  of  7\  -f-  Wr  for  a  bow  girder  built  in  at  both  ends,  subtending  an  angle  l.SO?  —  2<f>, 
and  carrying  a  single  load  W  at  a  distance  a  from  the  end  A. 

only  been  calculated  for  the  extreme  cases  likely  to  be  found  in  practice — viz.,  forl^T: 
CJ  —  1*25  (its  approximate  value  is  a  solid  circular  section)  and  for  El:  CJ  =  100, 
These  results  are  plotted  as  curves  in  Figs.  13  and  14,  and  for  intermediate  values  of 
the  ratio  the  moments  and  reactions  may  be  obtained  with  a  sufficient  degree  of 
accuracy  by  interpolation  from  these  curves. 

Owing  to  the  relatively  greater  variation  in  the  end  torques,  values  of  these  for 


24  A   STUDY   OF   THE   CIRCULAR-ABC   BOW-GIRDER 

a  series  of  values  of  El:  C'J  have  been  calculated,  and  are  plotted  in  Figs.  15  and  16. 
By  substitution  from  these  values  in  equations  (4),  (5),  (12),  and  (13),  the  values  of 


10 


4-0°  50° 

Va-lues  oF  CL 


FIG.  1<>. — Values  of  TK  ~  \\  r  for  a  bow  girder  built  in  at  both  ends,  subtending  an  angle  ISO0  —  '1$, 
ami  with  a  single  load  II'  at  a  distance  a  from  the  end  J. 

the  bending  and  twisting  moments,  and  of  the  deflections  at  any  point  of  the  girder, 
may  be  obtained. 

Special  Cases. 

Semicircular  Bow-Girder  with  Single  Load  W  in  any  Position. — Here  a  -\-  fi  = 
180°  ;  <£  —  0  ;  and  the  foregoing  equations  simplify.     The  values  of  the  various   con- 


CIRCULAR-ARC   GIRDER,   BUILT   IX   AT   TWO   EXDS        25 

stants  for  such  a  girder  have  been  calculated  for  the  case  where  El  =  1*25  CJ,  and  are 
given  in  Table  I. 

TABLE  I. 


a 

0° 

1  ?  I 

15° 

30° 
l^o 

45° 
135 

60° 
llo 

75° 

1  D< 

90° 

IL 
W 

1-00 

•990 

•940 

•870 

•764 

•640 

•500 

B, 
W 

o-o 

•0104 

•060 

•131 

•236 

•361 

•500 

Ma 

Wr 

o-o 

•239 

•128 

•542 

•590 

•571 

•500 

Mi 
Wr 

o-o 

•0200 

•0725 

•165 

•276 

•395 

•500 

T,, 
Wr 

o-o 

•0251 

•0662 

•115 

•155 

•181 

•182 

T,, 
Wr 

o-o 

•0118 

•0382 

•082 

•128 

•161 

•182 

In  the  particular  case  where  a  =  90°  =  T5708  radians  (i.e.,  weight  at  centre 
of  span)  from  symmetry 


Mn  =  Mh  =  -5  Wr 
Ta  =  T,, 


From  (10)  the  value  of  ~  at  the  weight  fa  =  ^j  is  given  by 


From  symmetry  tbis  equals  zero  ; 


and  in  this  case  both  Ma  and  Tn  are  independent  of  the  relative  values  of  El  and  CJ. 
The  curves  of  Fig.  17  and  18  show  respectively  the  values  of  the  bending  and  twisting 


26 


A   STUDY   OF   THE   CIRCULAR- ARC   BOW-GIRDER 


moments  at  each  section  of  a  semicircular  girder  clue  to  a  single  load  IT  at  any  distance 
a  (degrees)  from  one  end,  and  ordinates  of  the  envelopes  to  these  curves — shown  in 


CIECULAK-AKC   GIEDEE,   BUILT   IN   AT   TWO   ENDS        27 


0=  jo  fan /erf 


•t 

H 


a  g 


dotted  lines — give  the  maximum  positive  or  negative  moments  produced  at  any  point 
by  a  concentrated  rolling  load  of  this  magnitude. 


28 


A   STUDY   OF   THE   CIRCULAR- ABC   BCTW-GIRDEB 


Circular- Arc  Girder,  subtending  an  Angle  less  than  180°,  and  carrying  a  Single 
Weight  at  the  Centre  of  the  Span. — Let  2a  =  (-  —  2</>)  be  the  angle  subtended 
(Fig.  12).  The  moment  of  the  weight  about  AB  =  }Vr  (1  —  sin  </>),  and  as,  from 
symmetry,  Ma  =  Mb ;  Tu  =  Tb  ;  equation  (8)  becomes 


HV 


or 


also 


31  a  cos  d>  —  Ta  sin  d>  =  ——  (1  —  sin  <f>) 

2 

II V 


2  cos  </> 


(1  —  sin  </>)  -f  7',,  tan  </>, 


7?  =  It  = 
On  substituting  these  values  of  37,,  and  Ra,  equation  (10)  becomes 

~Tn-3  r/i  — sind.  .  r« 


d 


2  cose/) 


I—  si 


^ 


,"1 


From  symmetry  this  equals  zero,  and,  on  substituting  for  a  and  </>  and  equating 
to  zero,  the  value  of  Ta  is  obtained.  Except  in  a  semicircular  girder  (</>  =  0),  this 
value  depends  on  the  ratio  of  El  :  CJ.  The  following  values  have  been  calculated  for 
the  case  in  which  this  ratio  equals  1'25. 


</>° 

0° 

15° 

30° 

45° 

60° 

ua 

Wr 

•50 

•410 

•314 

•223 

•140 

Tn 
Wr 

•182 

•099 

•045 

•0157 

•0032 

Knowing  Ma  and  7',,,  tha  deflection  at  the  weight  may  IK;  obtained  by  substituting 
these  values  in  equation  (14),  p.  20. 

(16)     Circular-Arc  Bow-Girder,    Built  in  at  both  Ends,  with  Uniform  Loading  — 

ic  Ibs.  per  Unit  Length. 

Let  77  —  2<£  be  the  angle  subtended  by  the  arc  (Fig.  12).     The  total  load  =  UT 
(-  —  2$)  Ibs. 


The  centre  of  gravity  of  the  load  is  at  a  distance  from  the  line  of  supports  given 


(16) 


/• 


77    —    2</) 


CIBCULAE-AEC   GIBDEB,   BUILT   IX   AT   BOTH   ENDS       29 

Let  Ma,  Mb,  Tn,  Tb,  have  the  same  meanings  as  before      Then,  from  symmetry, 
;  _  Mh)  Trt  =  Tb ;  and,  on  taking  moments  about  the  1 


Values    of     E I  :  C  J 
EIG.  19.-Values  of  M«  for  a  girder  with  uniform  loading,  subtending  an  angle  180°  - 


( 
2A/ft  cos  <£  -  2Trt  sin  </>  =  2«T-  jcos  </>  -^  - 


.     (17) 


30 


A   S'lTDY    OF   THE   CIRCULAK-ARC   BOW-GIEDKIJ 


Taking  the  origin  of  </>  at  the  supports, 

Me  =  J/,,  cos  9  —  liar  sin  0  +  Tn  sin  0  +  wr\l  —  cos  0)  l 

=  (Mn  -  in-2)  cos  0  -  (U(,r  -  Ta)  sin  d  +  ?rr2  . 
7*  =  Ta  cos  0  +  AV(1  —  cos  0)  —  3/,,  sin  0  —  i<va(0  —  sin 
=  (77rt  -  Iiar)  cos  0  -  (M,,  -  in3)  sin  0  +  Rnr  -  icr20 

If  the  girder  is  fixed  horizontally  at  the  ends, 


.     (18) 
•     (19) 


cos    l  ~ 


"  sn  (l  - 


0-3 


f  0-2 


0-1 


Se/77 


75. 


30 


m 


iraert-- 


40 


SO 


70 


/OO 


Values  or  £  I :  CJ 
FIG.  20. — Values  of  Ta  for  a  girder  with  uniform  loading  subtending  an  angle  18()c 

and,  on  substituting  for  Md  and  T0  from  (18)  and  (19),  this  gives 

(<!.'t\  . 


26V 


!  cos  0!  +  sin  0J  -  (/A,r  -  7'a)^  sin  0!  +  2,n  2sin 

_  r(Ta  —  Ear)el  sin  9l  —  (Ma  —  ?rr2){sin  0i  —  0t  cos  ^H 
V  L  +  27iV(l  —  cos  0J  —  2;n2(<91  —  sin  0J  J 


(20) 


Writing  0  for  ^  in  this  expression,  and  integrating  between  the  limits  6l  and  0 
we  have 


1  The  last  terms,  representing  the  moments  due  to  the  portion  of  the  load  between  A  and  6,  being 
obtained  as  at  the  beginning  of  §  (4). 


CIRCULAK-ARC   GIRDER,   BUILT   I1ST   AT   BOTH  ENDS       31 


10" 


20"  30° 


40°  SO" 

Values  of  (j> 


60" 


70° 


80' 


3iT 


FIG.  21. — Values  of  Ma  and  of  ra  in  a  girder  with  uniform  loading,  subtending  an  angle 

[180° -24*].     EI:CJ  =  1Q. 


,2     r 

yj  \(Ma  —  wr^sin  01  —  (Rar—Ta)(sm  6±  —  ^cos^)— 2w?-2  (cos^— : 

(Ta  —  Rar)  (sin  01—01  cos  ^)+  {(Ma—wi*)  (6l  sin  0x+2  cos  ^—5 


2CJ 


cos      - 


(21) 


From    symmetry  ^  is    zero   at    the   centre   of   the    span   where  6l  =~   —  ^,,    and 

A 

by  substituting  this  value  for  ^  in  (20),  and  by  also  substituting  for  Ma  its  value 

f  /7T  T  \  I 

it;r2|l  —  (^-  —  <^  _  — ^j  tan  </>  |  and  equating  to  zero,  the  value  of  Ta  may  be  obtained, 


A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


after  which  the  values  of  M6  and  Te  for  any  point  on  the  girder  may  be  obtained 
by  substitution  in  (18)  or  (19). 

The  values  of  Mlt,  Ta,  M0,  T0  have  been  calculated  from  the  foregoing  equations 
for  one-half  of  a  uniformly  loaded  girder  for  a  series  of  values  of  $,  and  of  6  for  each 
value  of  </>.  These  values  depend  slightly  on  the  relative  value  of  El  and  of  CJ , 
and  in  Figs.  19  and  20  values  of  M,t  and  of  T,,  are  plotted  for  a  series  of  values  of 
El :  CJ.  Fig.  21  shows  the  variation  of  M,,  and  of  Tn  with  tf>,  for  a  given  value  of 
El  :  CJ.  The  curves  of  this  figure  are  calculated  for  the  case  where  this  ratio  equals 


i-o 


•80 


•60 


N^<r 


p   'Zo 


&£ 


o^ 


^ 


20°          -5o°  40"          5"o°          Go" 

Values  of  6  measured  from  one  support 


80° 


FIG.  22. — Bending-inoiiiciit  diagrams  for  one-half  of  a  uniformly  loaded  circular- arc,  subtending 

an  angle  of  [180°  —  2<p]. 

10,  and  for  purposes  of  design  these  values  may  be  taken  as  sensibly  accurate  for  any 
likely  values  of  the  ratio. 

Figs.  22  and  23  show  respectively  the  bending  moment  M0,  and  the  twisting 
moment  T6  at  each  point  of  a  uniformly  loaded  bow  girder  subtending  an  arc  180— 2</> 
degrees. 

Special  Case. 

Semicircular  Girder  with  uniform  Load. — Here  $  =  0,  and  we  have  :  — 
Mlt  =  Mb  =  wr2  :  Ea  =  Rb  —     .  wr  : 


CIECULAE-AEC   GIRDER,   BUILT   IN   AT  BOTH   ENDS       33 


^  [(7'0  -  Rar)0i  sin  0, 


r  sn 


sn 


-  cos       ~    <n-(     -  sn 


(20') 


/l>«r)(sin  *i  ~  ^  cos  i>  ~  2 

(7Trt  —  Z»V)(sin  ^!  —  0J  cos  0X) 


—  sn 


•     (21') 


Values  of  d  measured  From  one  support . 


FIG.  23. — Twisting-moment  diagrams  for  one-half  of  a  uniformly  loaded  circular-arc  girder, 

subtending  an  angle  [180°  —  20] . 

Substituting  for  Ma  and  Ra  in  (20'),  writing  ~  for  0,  and  equating  to  zero,  gives 
Ta  =  ivr2  X  ?  (~  -  2)  =  -298wr2, 

77   \4  / 

and  on  substituting  in  (18)  and  (19) 

M6  =  wr\\  —  1-2728  sin  0), 
Te  =  W7\1'570S  —  1'2728  cos  0  —  0). 

1 


This  makes  M6  =  0  when  sin  0  = 


=  -7850  ;  *.«.,  when  6  =  51°43',  and 


1-2728 

makes  Te  =  0  when  0  =  22°40',  and  again  when  0  =  90°.  Affl  is  a  maximum  when 

~-~  =  0  ;  i.e.  when  cos  0  =  0,  and  therefore  at  the  supports.  T,?  is  a  maximum  when 

E.G.  D 


34 


A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


cW 


=  0,  i.e.  when  sin  6  =  '7850,  or  when  6  =  51°43'. 

7T 

Writing  61  =  ~  in  (21')  and  substituting  for   Ta  and  Rn,  the  deflection  at  the 

A 


4  P7272       -053~| 

^centre,-  _   ^.j   +26VJ    ' 


centre  is  given  by 


(17)     Circular-Arc  Bow-Girder,  Subtending  an  Angle  (180— 2(/>)°,  Built  in  at  the 
Ends  and  Carrying  a  Uniformly  Loaded  Platform. 

Let  w   Ib.   per   unit  area   be   the   load   on   the   platform    whose    area    will   be 

r2  f  ) 

-^  -j  TT  —  20  —  sin  20  \ .      Imagine   the   latter   to    be   divided   into  a  series  of  strips 
^  I  ) 

parallel  to  AB,  each  of  these  strips  transmitting  its  load  to  the  girder  at  its  ends.    The 
length  of  the  particular  strip  resting  on  the  girder  at  points  distant  6  from  A  and  B, 


FIG.  24. 


is  2r  cos  (6  +  0)  (Fig.  24).      If  this  strip  covers  a  length  S.s  =  rSd  of  the  girder,  its 
width;is  r&6  cos  (6  +  </>),  and  the  load  on  it  is  2zw2  cos2  (9  +  </>)§#. 

Its  moment  about  AB  =  2«;r3  cos2  (0  +</>){  sin  (9  +  (/>)—  sin  <£}  80, 


/.  Moment  of  whole  load,  about  ^1£  = 


~* 

=  Zivr* 

Jo 


+^){sin(6'  +  </>)-  si 


0, 

—  2     —  sin 


Since,  from  symmetry,  Ma  =  Mb  ;  Ta  =  Tb;  it  follows  that 

Af0  cos  0  -  Ta  sin  0  =  wi*  [^  -  ^  (rr  -  20  -  sin  20)} 
.,  Ma  =  ^{2^*  -  ^  (W  -  20  -  sin  20)}  +  Ta  tan  0. 
Again,  since  the  total  load  is 


'   I  cos2  (0  +  0>70 

0 


CIRCULAR-ARC   BOW-GIRDER,   SUBTENDING  AN  ANGLE     35 


.-.  Ra  =  Kb  =  -  -\TT  —  20  —  sin 20 \ . 
4    (  r  j 

The  bending  and  twisting  moments  at  a  point  xr  distant  Ql  from  OA  are  given  by 

r 

Mei  =  Ma  cos  0X  —  (Kar  —  Ta)  sin  0X  +      wr3  cos2  (0  +  0)  sin  ft  —  0)  dO. 


T6i  =  (Ta  —  Rar)  cos  0  —  Ma  sin  6  +  Rar  —     ™r3  cos2  (0  +  <j>)  {1  —  cos  (ft  -  0)}d0, 


the  last  term  in  each  case  representing  the  moment,  bending  or  twisting,  about  the 
point  xl  (Fig.  24),  of  the  load  between  A  and  xr 

On  integrating  these  terms  and  writing  d  for  dlt  the  general  expressions  for  M6 
and  Tg  become 

~  l){cos  6  —  si 


a      fi>        -r\    •    a  . 
M.  =  Ma  cos  0  -  (Rar-l  J  sm  ^  +  — 


=  (a-     ar  cos 
—  M  sin  ^ 


ivr 


6        .     . 
-  +  sm(9 


, 

[_  _  (i  _  cos  (9)  — 


sin  2<i  ,,  ,,,, 

^-.(l.—  cos  Qy 
b 


(22; 
(23) 


As  before,  if  the  girder  be  fixed  horizontally  at  the  ends 
dy\      -  »'' 


and,  on  substituting  the  foregoing  values  of  Me  and  Te  and  integrating,  this  gives  the 

value  of  -jTj  at  any  point  61.     Thus 
du 


fjft  cos  9i  -\-  sin  0X)  —  (Rar  —  T^O^  sin  8^ 

I  /7  \ 

.sin  6.  (     —  sin20{3-f  0,}     —  03  cos 

/•^•O   I  1   \   O  / 

-3! 


(Ta-Rar)dl  sin  ^-. 

7  . 


sin2(/>) 


?-'J 

~t~  i^n 


-     cos 

cos2</>  .  sin2<^  .  cos  0]  cos  20 
--  -     —  — 


0 


cos 


.    .  sin2<f>      2\    .   sin 
0  +  —  g-f  -  gj  +  —9 


—  sn     i  +  cos    i  - 


(24) 


From  symmetry  the  slope  is  zero  at  the  centre  of  the  beam  where  #1  ;=  ^  —  0,  and, 

on  substituting  this  value  for  #1  in  (24),  and  also  substituting  the  values  of  Ma  and  Ra 
as  given  on  pp.  34  and  35,  and  equating  to  zero,  the  value  of  Ta  may  be  obtained. 
E.g.,  Semicircular  Girder  (0  =  0). 

D  2 


36  A  STUDY   OF   THE   CIECULAE-AEC   BOW-GIEDEE 

In  this  case,  on  putting  <£  =  0  in  (24) 

Ma{0icos  0i  +  sin  0i}  -  (Rar  —  Ta)0i  sin  ( 


2EI 


+ 


'in3'  (7  /4  \  ^ 

.«/ll*          f\  rt     I  *      "         t\         \        f\      \ 

+  ^sm0i-cos01^sm0i+0ij; 


2CJ 


(!Tfl  -JR^  sin 


—    i  cos 


+  27?ftr(l  -  cos  0i)  -  in-3-!  0i  -  sin  0i  (-^   -  ~  cos  0i 


(24') 


At  the  centre,  where  0i  =  -.,  the  slope  is  zero,  and  Mtl  =  -g- ',  Ka  —  wr    •  4- 

wr*  Tl        (IT  ^_  Ta  \TT       7"|    ,    n-r5   f  /  T«  _  TT\  TT  __  j.    ,    lg"|  _  0 
•  •  2£7  L3     "  \4  _  icrsJ  2  """  9j  ^  2C'J  L  Ur3       4/2      3 


It  follows  that,  on  substituting  in  (22)  and  (23) 

o    fl   4-  Sin2  0  r,f\r,A      '        a 

Me  =  wr3  \  '7074  sm  0 

(  o 

(sin  0cos  0  .  TT 


=  ivr 


.  T  n 

+  T  —  '7074  cos  0  —  ^ 

I  D  4  « 


The  deflection  at  any  point  0i  is  obtained  by  writing  0i  =  0  in  (24')  and  integrating 
between  the  limits  0i  and  0.     Thus, 


ZCJ 


Mu6j  sin  61  -  (Rar  -  Ta)  (sin  8±  -  ^cos  0X) 

4.  ^{10  -  10  cos  0X  -  2  sin2  0i  -  30X  sin  81} 

v      \  '  . 

f  (Ta  -  Rar)  (sin  0i  -  0i  cos  00  +  2Rar  (0i  -  sin  00 
+  Ma  (0i  sin  0!  +  2  cos  0i  -  2) 


9    I    2 

77 


16cos  0i  -  16 


+  30.  sin 


in  0l\ 
J  —  I 


(25) 


At  the  centre,  where  0i  =  „ 


^/centre  — 


-  "?t  P1815   ,   -01211 

T  i .  JH       cy  J 


GIEDEE  WITH   UNSYMMETEICAL   LOADING  37 

(18)     Girder  with  Unsymmetrical  Loading. 

Where  the  loading  of  a  girder  does  not  admit  of  being  represented  by  a  simple 
trigonometrical  expression,  or  where  the  girder  is  not  of  uniform  cross  section  through- 
out its  length,  a  solution  is  most  readily  obtained  by  dividing  the  load,  including  the 
dead  load  due  to  the  girder  itself,  into  a  series  of  comparatively  short  lengths,  and  by 
calculating  the  moments  due  to  each  of  these  portions  of  the  load  separately,  by  an 
application  of  the  reasoning  and  results  of  §  (15).  In  practice  a  first  approximation 
would  be  obtained  by  assuming  a  likely  value  for  the  cross  section  and  weights  at  each 
point,  and  by  then  applying  these  results.  A  second  approximation  would  then  be 
made  taking  into  account  the  weight  of  the  girder  calculated  from  the  sections  found 
necessary  by  the  first  approximation,  and  this  would  in  the  majority  of  cases  give 
results  sufficiently  near  for  all  practical  purposes. 

(19)     Bow-Girder  Built  in  at  the  Ends  and  Resting  on  Intermediate  Supports. 

Assuming  all  the  supports  to  be  at  the  same  level,  the  reactions  of  the  intermediate 
supports  may  be  most  readily  obtained  by  expressing  the  fact  that  the  upward 
deflections  at  these  supports  caused  by  their  reactions,  are  equal  to  the  downward 
deflections  produced  at  the  same  points  by  the  loading. 


(20)     Girder  with  Uniform  Loading  and  Central  Support. 

Let  P  be  the  reaction  of  this  support.  Let  180  —  2$,  or  2a,  be  the  angle  sub- 
tended by  the  arc  of  the  girder. 

The  upward  deflection  at  the  centre  due  to  the  reaction  is  given  by  equation  (14), 
in  which  W  '=  P,  and  in  which  Ma  and  Ta  have  the  values  given  by  the  curves  of 
Figs.  13  —  16,  for  the  corresponding  value  of  a  or  (90°  —  </>).  The  downward  deflection 
at  the  centre  due  to  the  load  is  obtained  by  substituting  a  for  dit  and  by  substituting 
the  corresponding  values  of  Ma  and  T'a  as  given  by  the  curves  in  Figs.  19  —  21,  in 
equation  (21). 

E.g.,  ct  =  90°  ;  </>  =  0  (semicircular  girder). 

The  upward  deflection  at  centre 

=  m.  [I  -e^o-182)]  +  ^  [oi82  -  -5oo)  +  1  -  1  +  1  (i  _  2)] 

.  p,.3  f-4674       -03821 
L  2A7         2CV  J  ' 

The  downward  deflection  at  the  centre,  due  to  the  loading 

-7272    , 


and  on  equating  these 

-727207+  -053EI 


_ 

r 


'4674C'Jr 


The  value  of  this  depends  slightly  on  the  ratio  of  El  to  CJ.     Taking  this  ratio  as 
T25,  gives 

'7928 


A   STUDY   OF   THE   CIECULAR-AEC   BOW-GIEDEE 

Again,  7?a  -f  Rb  +  P  =  trier 

•'•  ^a  =  #6  =  -^-  \TT  — 

Also 


=  -801/cr. 


.17,,  +  Mb  =  2«T2  -  Pr 

=  -46/n-2 

/.  .1/H  =  .V6  =  -23«v2. 
The  value  of  Ta  is  the  difference  between  the  values  produced  by  the  load  and  by 


20° 


80' 


90° 


30°  4-0  50°  60'  70° 

Ka/i/es  of  8  measured  from  the  end  support:. 

FIG.  25. — Bending  moments  in  a  uniformly  loaded  circular-arc  built  in  at  the  ends  and  having 

a  central  support.     (Full-line  curve.) 

the  upward  reaction  P.       The  first  of  these  is  -Z98wr*  (Fig.  20) ;  the  second  is  -182/V 
(Fig.  16). 

.-.  Ta  =  {-298  -  (-182  x  1-54)}  «r2 
=  -018«T2. 

This  value  may  be  obtained  alternatively  by  substituting  the  foregoing  values  of 
Ma  and  of  El(  in  equation  (20)  with  Oi  =  „>  and  by  equating  to  zero. 

£ 

The  values  of  Mg  and  of  Te  at  any  point  between  the  end  and  the  support  and 
distant  d  from  the  end  then  become,  on  substituting  in  equations  (18)  and  (19) 
Mg  =  u-r2{l  —  '11  cos  0  —  '783  sin  d}, 
Te  =  MT2{-801  —  -783  cos  6  +  '77  sin  d—  d}. 


GIEDEE  WITH  UXIFOEM  LOADING  AND  CENTRAL  SUPPOKT     39 


If  El :  CJ  =  10  the  values  of  the  end  moments  and  reactions  become  P  =  T47  wr ; 
Ra  =  Hb  =  -885MT ;  M(t  =  Mb  =  '265«T2  ;   Ta=Tb  =  -OSOirr2,  and  equations  (18)  and 

(19)  become 

Me  =  wr*{l  —  '735  cos  0  —  "805  sin  d\ 

To  =  HT2{-835  —  -805  cos  9  +  '735  sin  6  —  9}. 
Figs.  25  and  26  show  the  bending  and  twisting  moments  at  each  section  of  one- 


20°  30°  40"  50  60°  70°  80' 

Values    oF    Q   measured  From    the  end    Support. 

FIG.  26. — Twisting  moments  in  a  uniformly  loaded  circular-arc  built  in  at  the  ends  and  having 

a  central  support.     (Full-line  curve.) 

half  of  such  a  girder  with  a  central  support  and  with  El  -r-  CJ  =  1'25,  while  for  com- 
parison the  moments  with  the  same  loading  but  without  the  central  support  are  shown 
by  the  dotted  line  curves  on  the  same  diagrams. 

Where  the  girder  subtends  an  angle  less  than  180°,  the  problem  may  be  solved  in 
an  exactly  similar  manner  by  making  use  of  the  requisite  relationships  from  the  fore- 
going curves. 


50 


40  A   STUDY   OF   THE   CIKCULAK-AKC   BOW-GIEDEE 

(21 )      Circular- Arc  Girder,  built  in  at  the  Ends,  with  Uniform  Loading,  and  with 
two  Symmetrical  Intermediate  Supports. 

Let  the  angle  subtended  by  the  girder  be  (180— 2<£)°,  and  let  the  supports  (at  C  and 

D,  Fig.  27)  be  distant  y  from  each  end.     Let  the  upward  reaction  at  each  support 

-  P.     Let  Ma",  Ta",  RJ'  represent  such  end  conditions  at  A  as  would  be  produced  by 

these  two  reactions  alone,  and  let  Ma',  Ta',  Ra'  -represent  such  end  conditions  as  would 

be  produced  by  the  load  alone,  with  supports  removed. 

Under  these  conditions  the  downward  deflection  at  C  and  D  due  to  the  loading 
would  be,  by  equation  (21) 


2/v  = 


-(^«V—  r((')(smy-7cosy)  — 


•ACJ 


~(Ta'  -Ra'r)  (sin  y  -  7  cos  7)  +  (Ma'  -  «T2)  {  y  sin  y  + 

2 

2  cos  7  -  2}  +  2/4'r(y  -  smy)-2«T2(^-+cos7-  1) 

.  —  — 


-«] 


(26) 


where  Ra'  =  utri-—  </>J ,  and  A// and  Ta'  for  the  particular  value  of  $  obtaining  in 

the  girder,  are  given  by  the  curves  of  Figs.  19—21. 

The  upward  deflection  at  C  and,  from  symmetry,  at  D,  due  to  the  two  upward 
forces  P  is  obtained  by  substituting  y  for  61  in  equation  (12),  which  becomes 


~  |Vtt"  y  sin  y  -  (Ru"r  -  Ta")  (sin  y  -  y  cos  y)] 

r*_  r(Ta"  -  Ra"r)  (sin  y  -  y  cos  7)  +  2flrt'V  (y  -  sin  7)  "1 
*~  1CJ  L  +  Ma"  (y  sin  y  +  2  cos  y-  2) 


(27) 


The  values  of  Ma",  Ra",  Ta"  for  use  in  this  expression  are  the  sum  of  the  corresponding 
values  produced  by  each  of  the  two  forces  P  acting  at  points  distant  y  from  A  and  from 
B,  and  may  evidently  be  obtained  by  adding  the  values  of  Ma  and  Mb,  Ra  and  R/t,  Ta 
and  Tb,  as  obtained  from  the  curves  of  Figs.  13 — 16  for  a  girder  having  the  correct 
value  of  <£,  and  having  the  force  P  at  y  from  A. 

On  substituting  these  values,  each  of  which  is  given  in  terms  of  P,  in  equation 


CIRCULAR- ARC    GIRDER,   BUILT   IX   AT   THE   EXDS         41 

(27)  and  equating  to  (26),  the  resultant  expression  contains  P  as  the  only  unknown  and 
enables  this  to  be  calculated. 

E.g.,  Semicircular  girder  with  uniform  loading  and  with  two  piers  at  60°  from 
the  ends  of  the  span  (</>  =  0  ;  y=  60°). 

From  Figs.  19  and  20  the  values  of  Mar,  and  Ta'  for  substitution  in  equation  (26) 
are  Ma'  =  wr2;  Tn'  —  '298/rr2 ;  while  74'  =  1'5708/rr,  and,  on  substituting,  the  down- 
ward deflection  at  the  supports  (y  =  60°)  is  given  by 

,  P564   ,    -0371 


The  values  of  Ma",  Ttt",  and  74"  for  substitution  in  (27)  are,  from  Figs.  13,  14,  15 
and  16 

M,,"  =  (.17,,  +  Mb)t  =0>  y  =  6rp  =  (-588  +  -278)  I'r  =  '8Q6Pr. 
Ta"  =(Ta+  Tb\  =  o,  y  =  GO>  =  ('156  +  -127)  Pr  =  '2837V. 
R  "  —  P 

•"'a    —  •*  > 

and,  on  making  these  substitutions, 

v  s  P539   .    '0351 

//'"  '    L2A7  "*"  2Gvd  * 

Equating  these  two  expressions  for  //00°  gives 

r.564(7j  +  -o37/';n 

L-539CV7  +  -035 Elj  ' 

and  taking  El  =  1'256'J",  this  makes  P  =  I'OS^r. 
The  reactions  at  A  and  73  are  then  given  by 

74  =  74  =  7C  -  74"  =  in-  (j  -  1-05Y=  -521/rr. 

\ij  / 

while  the  moments  Ma  and  A//;  are  given  by 

,17,  =  Mn  =  Ma'  —  Mfl"  =  wrz  (1  -  "866  X  1'05)  =  -091«-/-2. 
The  torques  Ta  and  Tb  are  given  by 

Tb  =  Ta  =  Ta'  -  Ta"  =  jrr2{-298  -  '283  X  1'05}  =  '001/rr2. 

The  state  of  affairs  at  any  point  on  the  girder  is  thus  given  by  the  relations 
(equations  (18)  and  (19) )  : — Between  A  and  C — 

Me  =  Ma  cos  0  -  (Rar  -  T,()  sin  9  +  WIA  (1  -  cos  6) 

=  wi*  jl  -  '909  cos  6  -  -520  sin  9\ 

Te  =(Ta  -  Rar)  cos  B  +  R,tr  -  Ma  sin  6  -  icr2  (d  -  sin  0) 
=  wr2  {-521  -  '520  cos  6  +  "909  sin  0-0} 

Between  (7  and  the  centre  (0  being  measured  from  OA) — 

Me  =  Ma  cos  0  —  (74?-  —  Ta)  sin  0  +  wr*  (1  —  cos  0)  —  Pr  sin  (0  —  60°) 

=  wr*  1 1  —  1-045  sin  0 1 
Te  =  (Ta  -  74r)cos  0  +  /4>-  -  ,l/ft  sin  0  -  /rr'2(0  -  sin  0)  +  7J;-  j  1  -  cos  (0  -  60°)  j 

=  «r2  1 1-571  —  1-045  cos  0  —  0 ; . 

Fig.  28  shows  the  bending  and  twisting  moment  diagrams  for  such  a  girder,  while 
for  purposes  of  comparison  these  have  also  been  drawn  as  dotted  line  curves  on  Figs.  25 
and  26.  From  these  it  appears  that  the  maximum  values  of  the  moments  with  and 
without  supports  have  the  following  ratios,  the  bending  and  twisting  moments  for  the 
span  without  intermediate  supports  being  taken  as  unity. 


42  A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


Number  of   Intermediate  Supports. 

none 

one  at  centre 

two  at  60° 

Maximum  bending  moment 

1-0 

•26 

•09 

Maximum  twisting  moment 

1-0 

•11 

•035 

•10 


o 


Ol 


90 


FIG.  28. — Diagrams  of  bending  and  twisting  moments  for  uniformly  loaded  semicircular  girder, 
with  two  intermediate  supports,  distant  60°  from  each  end. 


The  following  table  shows  how  the  fixing  moments  and  reactions  vary  with  the 
ratio  of  El :  CJ  in  the  foregoing  example. 


El 
CJ 

P 

UT 

Ra_ 

•ivr 

M± 

wr- 

T» 

«•/•- 

1-25 

1-05 

•521 

•091 

•001 

100 

1-06 

•511 

•081 

-•002 

SEMICIRCULAR   GIRDER,   BUILT   IN  AT   THE   ENDS        43 

From  these  figures  it  appears  that  a  considerable  change  in  this  ratio  has  very 
little  effect  on  the  magnitude  of  these  moments. 

Semicircular  Girder  with  uniform  Loading  and  with  two  Piers  at  45°  from  Ends  of 

Span. 

In  this  case,  the  end  constants  and  pier  reactions  for  EI=1'25  CJ  become 
P  =  l-460?tr  ;  Ma=    -  -QSlwr2  ; 

Ea  =  Rb  =  -llltcr        •  Ta  =  -010/rr2. 

As  before,  between  A  and  C 

Me  =  Ma  cos  e  -  (Rar  -  Tn)  sin  d  +  «v2  (1  —  cos  6), 
Te=(Ta-  Rar)  cos  6  +  Rar  -  Ma  sin  0  -  wr*  (d  -  sin  6), 

while  between  C  and  the  centre 

Me  =Ma  cos  9  —  (Ear  -  Ta)  sin  0  -  Pr  sin  (0  —  45°)  +  wrz  (1  -  cos  0), 

Te  =  (Ta  -  Rar)  cos  0  +  Rur  -  Ma  sin  0  +  Pr  {1  -  cos  (0  -  45°)  }  -  wr*  (0  -  sin  0). 

(22)    Semicircular    Girder,  built  in  at  the  Ends,  with  Uniform  Loading,  and  with 

three  Intermediate  Supports. 

Let  the  supports  be  arranged  symmetrically,  P1  and  P2  being  the  reactions  at  the 
outer  supports  and  Q  that  at  the  central  support.  These  reactions  may  be  obtained 
by  expressing  the  facts  (1)  that  the  downward  deflection  at  the  centre  due  to  the 
loading  is  equal  to  the  sum  of  the  upward  deflections  at  the  centre  due  to  the  forces 
JPlt  P2,  and  Q,  in  their  respective  positions  ;  and  (2)  that  the  downward  deflection  at 
PI  due  to  the  loading  is  equal  to  the  upward  deflection  at  this  point  due  to  forces 
P15  P2,  and  Q  ;  thus  if,  for  example,  Px  and  P2  are  each  at  45°  from  the  ends,  we 
have  — 

Downward  deflection  at  Q  due  to  loading 

J-7272   ,    -053) 
—  j/i-r*  -{  _  _i_  -    _  '- 

\<2EI  +  26VJ- 
Downward  deflection  at  Pl  or  P2  due  to  loading 


I 

2A7    h  26V/' 

these  values  being  obtained  from  equation  (21')  by  substituting  the  values  of  0,  viz.,  90° 
and  45°,  and  of  Ma  and  Ta  from  Figs.  19  and  20. 

Again,  the  upward  deflection  at  Q  due  to  force  Q 
.  f-4674    ,    •0882~l 
~'~ 


from  ^14^  and  FiSs-  13  and 
and  the  upward  deflection  at  Q  due  to  the  two  forces  Pl  and  P2(  =  P 

from  13  and  Fis-  13  and 


L 


Also  the  upward  deflection  at  Pl  due  to  force  Px 

=  Pt*  L~2El  +  2C7J    from  ^14^  and  Figs'  1B  and  14 
while  the  upward  deflection  at  P1  due  to  P2 

=  Pr3  +  from  (13)  and  Figs.  13  and  14, 


44  A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 

and  the  upward  deflection  at  Pl  due  to  force  Q 

and  Figs.  13  and  14. 

Collecting  and  equating  deflections  at  the  same  points  gives 

UT(-7272CJ  +  '053EI)  =  Q('4674CJ  +  -0382EZ)  +  ZJ(-422(X'</  +  '0594EZ), 
io<-3928CJ  +  -0218EZ)  =  #(-2297C'«/+  -015EZ)  +  P('271QCJ +  -QUQEI), 

where  P  =  Pj  =  P2- 

If  El  =  T25GV,  the  solution  of  this  gives 

0  =  -74wr  ;     P  =  -83«r. 


•06 


Values  of  6  measured  From  one  end  of  Girder 

FIG.  29. — Bending  and  twisting  moment  diagrams  for  one-half  of  a  uniformly  loaded  semicircular 
girder  with  three  intei mediate  supports  at  45°. 

From  this 


Also 


=  '37  MT. 

Mcl  +  Mb  =  2/n-2  —  2/Y  sin  45°  —  Qr 

=  -088/rr2 
.'.  Mu  —  Mb  =  -044/n-2 

while  '1\  (from  Figs.  15  and  16) 

=  -298HT2  —  '112P;-  —  -182Qr  —  '083Pr 

=  ('298  —  -297)  wr* 
=  -0010  jo-8. 

Fig.  29  shows  the  bending  and  twisting  moment  diagrams  for  one-half  of  this 
girder,  and  a  comparison  of  these  diagrams  with  those  of  Figs.  25  and  26  indicates  to 
what  extent  the  maximum  moments  are  reduced  by  the  addition  of  the  third 
support. 


45 


(23)    Effect  of  Depression  of  Supports. 

Where  a  bow-girder  is  used  to  support  the  circle  of  a  theatre,  intermediate 
supports  are  often  provided  by  cantilevers  built  into  the  rear  walls  of  the  theatre.  If 
erected  so  that  under  no  load  the  ends  of  these  are  level  with  the  end  supports  of  the 
bow  girder,  their  deflection  under  load  will  reduce  the  supporting  pressure  to  a  value 
below  that  obtaining  with  rigid  supports,  will  increase  the  end  reactions,  and,  generally 
speaking,  will  increase  the  average  bending  and  twisting  moment  over  the  whole 
girder. 

If  P  be  the  end  load  on  a  given  cantilever,  its  deflection  at  the  free  end  is 
proportional  to  P  and  is  equal  to  kP  where  k  depends  on  the  dimensions  of  the 
cantilever.  For  example,  if  of  uniform  section,  of  moment  of  inertia  1',  and  of  length 

73 

/  l-  — 


The  actual  deflection  under  load  of  the  bow-girder  at  this  point  is  thus  kP,  and  if  y 
would  be  its  deflection  with  the  support  removed,  the  upward  deflection  due  to  the 
upward  force  P  is  equal  to  y  —  kP. 

Expressing  y  in  terms  of  the  load  on  the  girder,  and  expressing  the  upward 
deflection  due  to  P  in  terms  of  P  as  in  §§  20,  21,  and  22,  and  equating  this  to 
y  —  kP,  the  pressure  P  on  the  support  is  obtained  in  terms  of  the  load  as  in  the 
examples  of  the  preceding  articles. 

E.g.,  Semicircular  Girder  with  Uniform  Loading,  built  in  at  the  Ends  and  Sup- 
ported at  the  Centre  by  the  End  of  a  Cantilever. 


Deflection   at  support,  with  support)  _      4  P7272 

removed         .  .)  "  L  1EI  "•    2G'JJ 

Actual  deflection  at  support       .         .    =  kP 

Upward  deflection  at  centre  due   to)  _  _  p  >3  F'4674    ,    •0882"! 
force  P I  :  "          "h         ~ 


p.  34 
p.  37 


L  2EI    '    2GV 


/.  P  =  wr 


•05  3"| 
2GVJ 


•7272  +  -053 


•4674 


Thus,  for  example,  if  the  cantilever  be  of  uniform  section,  of  moment  of  inertia  /' 

I3 
and  of  length  /,  so  that  k  =  Q7,  „  this  becomes 


•7  272  + -053^ 


P  —  wr 


The  following  table  shows  how,  in  the  case  where  I  =  r  and  7  =  I',  the  yielding 
of  this  support  would  modify  the  end  moments  and  reactions  as  compared  with  those 
experienced  with  a  rigid  support  or  with  a  cantilever  so  erected  and  designed  as  to 
deflect  under  load  to  the  level  of  the  end  supports.  These  figures  apply  to  the  case 
where  El  =  10  CJ. 


46 


A   STUDY    OF   THE   CIECTJLAE-AEC   BOW-GIRDER 


/' 

«•/• 

R,i 
tor 

M* 

,772 

Ta 

wri 

Eigid  support 

1-47 

•835 

•265 

•030 

Elastic  support  . 

•828 

1-157 

•586 

•147 

No  central  support 

— 

1-571 

1-00 

•298 

It  will  be  noted  that,  due  to  yielding  of  the  support,  the   end  moments  (and 
approximately  also  the  average  moments)  in  the  girder  are  increased  roughly  in  the 


FIG.  30. 

proportion  in  which  the  value  of  P  is  diminished.  Owing  to  the  comparative  shortness 
of  the  cantilever  it  will  generally  be  more  economical  to  design  this  so  as  to  take  the 
load  P  corresponding  to  a  rigid  support,  and  to  erect  this  with  sufficient  camber  to 
allow  its  deflection  under  this  load  to  bring  it  down  to  the  level  of  the  end  supports. 
The  same  reasoning  in  general  holds  however  many  intermediate  supports  may  be 
used. 

(24)  Compound  Bow-Girder. 

The  state  of  equilibrium  of  a  compound  bow-girder  of  the  type  illustrated  in 
Fig.  30,  may  be  obtained  by  an  application  of  the  methods  used  for  the  simpler  forms. 
In  the  case  shown,  with  intermediate  supports  at  the  points  of  inflexion  B,  C,  E  and 
F,  the  whole  of  the  reactions  and  the  end  moments  Ma,  Mg,  Ta,  Tg,  are  unknown. 
From  symmetry,  however,  with  uniform  loading  Mg  =  Mu;  Tg=Ta;  lif=lib; 
Re  =  7i,, ;  R,j  =  Ra,  so  that  in  effect  the  only  unknowns  are  Ma,  Ta,  liu,  H,,,  Rc. 

Knowing  the  radii  i\,  r2,  r3,  and  the  angles  dv  62,  03,  the  total  load  on  the  girder, 
and  the  position  of  its  centre  of  gravity,  may  readily  be  obtained  as  in  art.  16,  p.  28. 


COMPOUND   BOW-GIKDER  47 

Calling  wl  the  load,  let  x  be  the  distance  of  its  centre  of  gravity  from  the  line  joining 
AG,  and  let  xl  and  x2  be  the  distances  of  supports  B  and  C  from  this  line. 
Then  taking  moments  about  A  G  gives 


irl 
Again  —  ~Ra  +  Rb-\-  Rc, 

so  that  if  Rb  and  Rc  are  known,  Ra  and  Ta  may  be  deduced  from  these  equations. 
This  leaves  in  effect  three  unknowns,  Ma,  Rb  and  R0,  and  in  order  to  determine  these, 
three  further  equations  are  necessary. 

These  are  to  be  obtained  as  follows  :— 

(1)  Span  A  B.  —  Write  down  the  expressions  for  the  slope  and  deflection  at  B  in 
terms  of  Ra,  Ma,  and  Ta.     These  are  the  same  as  equations  (20)  and  (21),  pp.  30  and 
31,  with  i\  taking  the  place  of  r.     Equating  the  deflection  at  B  to  xero  gives  the  first 
of  the  required  relationships. 

(2)  Determine  values  of  Mb  and  Tb  from  equations  (18)  and  (19),  p.  30,  in  terms 
of  Ra,  Ma,  and  Tu. 

(8)  Span  EC.  —  Obtain  the  slope  and  deflection  at  C  in  terms  of  Mb,  Tb,  Ra  and 
Rb,  and  of  the  slope  at  B.  Equating  the  deflection  at  C  to  zero  gives  the  second  of  the 
required  relationships. 

(4)  From  equations  (18)  and  (19)  determine  3/c  and  Tc. 

(5)  Span  CD.  —  Obtain  the  slope  at  D  and  equate  to  zero.     This  gives  the  third 
relationship. 

(25)  Shear  Force  at  a  given  Section. 

The  vertical  shear  force  at  any  section  of  a  bow-girder  is  the  same  as  would 
be  experienced  at  the  corresponding  section  of  a  straight  girder  subject  to  the 
same  loading  and  to  the  same  reactions.  Thus,  between  an  end  support  —  reaction  Ra 
—  and  the  first  concentrated  load  Wv  the  shear  force  is  constant,  except  for  the  weight 
of  the  girder  itself,  and  equal  to  Ra.  Between  this  load  and  a  second  load  W2,  the 
reaction  is  Ra  —  W\. 

In  the  case  of  a  uniformly  loaded  girder,  carrying  w  Ibs.  per  foot  run,  the  shear  force 
at  a  distance  x,  measured  along  the  arc,  from  the  support  A  is  Ra  —  icx  for  all  points 
between  the  end  and  any  intermediate  support.  If  there  be  an  intermediate  support 
at  a  distance  x±  from  the  end  A,  and  if  its  reaction  be  Px,  the  shear  force  at  a  point 
distant  x  from  A,  between  this  intermediate  support  and  any  third  support,  is  given 

by 

Ra  +  1\  -  wx 
and  so  on. 

(26)  Experimental  Verification  of  Formulae. 

In  order  to  verify  the  formulae  of  this  chapter  by  experiment,  measurements  of 
deflection  have  been  made  by  the  authors  on  a  series  of  bow-girders  fixed  at  one  or 
both  ends  and  loaded  either  by  single  concentrated  loads  or  by  a  uniform  load.  Some 
of  these  girders  were  of  circular  section,  others  of  angle  section.  Values  of  El  and  of 
CJ  were  obtained  by  deflection  and  torsion  experiments  on  straight  lengths  of  the  same 
sections,  and  these  values  have  bsen  adopted  in  the  calculations. 


48  A   STUDY   OF   THE   CIBCULAK-ABC   BOW-GIRDER 

The  following  are  the  results  of  the  experiments  :— 

TABLE  II. 


Series 

Type  of 

Conditions. 

Angle 
subtended 

Deflection  (ins.). 

section. 

by  arc. 

Measured. 

Calculated. 

a 

Circular 

Circular  arc  cantilever 

90° 

1-469 

1-475 

with  weight  at  free 

135° 

4-475 

4-475 

end 

b 

» 

Ditto   with    uniform 

90° 

•510 

•502 

loading 

c 

» 

Semicircular  bow  gir- 

a  =  30° 

•043 

•043 

derfixed  at  endswith 

„       45° 

•117 

•115 

single  load  at  a  from 

„       60° 

•202 

•204 

one  end  —  deflection 

„      90° 

•307 

•307 

at  weight 

d 

» 

Circular  arc  girder  with 

120° 

•075 

•074 

single    weight    at 

centre 

e 

)J 

Ditto   with    uniform 

180° 

•310 

•306 

loading 

f 

Angle 

Circular     arc     girder 

90° 

•on 

•012 

with  weight  at  centre 

180° 

•124 

•116 

!l 

Angle 

Semicircular  bow'gir-  ( 
der  with  single  load  ) 

Deflection  at 
weight 

•036 

•032 

at    45°    from    one] 

end 

Deflection  at 

•OG8 

•072 

centre 

From  these  figures  it  appears  that  there  is  a  very   close   agreement   between 
experimental  and  calculated  values  in  every  case. 

(27)  Non-circular  Sections. 

The  foregoing  formulae  are  of  general  application  to  a  beam  of  any  section  of  which 
the  El  and  CJ  are  known.     The  former  of  these  products  is  usually  known  or  can  be 


NON-CIRCTJLAK  SECTIONS  49 

determined  by  calculation  with  a  close  degree  of  approximation  for  any  commercial 
section.  While  the  geometrical  polar  moment  of  inertia  J  of  any  section  may  also  be 
calculated,  the  product  of  this  J  and  the  shear  modulus  C  of  the  material  does  not,  how- 
ever, give  the  effective  value  of  CJ  for  use  in  these  formulae,  except  in  the  case  of  circular 
sections.  The  reason  for  this  and  the  question  of  the  effective  value  of  J  for  non-circular 
sections  is  considered  in  some  detail  in  the  following  chapter. 


B.O. 


CHAPTER   III 


(28)  The  Torsional  Rigidity  of  Non-Circular  Sections. 

ON  the  assumptions  that  the  displacement  of  every  point  in  a  section  under 
torsion  is  proportional  to  its  distance  from  the  centroid  of  the  section,  and  that  a 

section  originally  plane  re- 
mains plane  after  straining, 
the  angle  of  twist  of  a 
straight  member  of  length  / 
is  given  by 


TL 

CJ 


(28) 


;jl 


where  J  is  the  polar  moment 
of  inertia  of  the  section,  as 
deduced  from  its  geometrical 
properties. 

If  the  section  is  circular, 
these  assumptions  are  fully 
justified  by  experiment  so 

long  as  the  stresses  involved  are  within  the  elastic  limit  of  the  material. 

But  this  is  not  the  case  for  any  but  a  circular  section.     In  any  other  section 

radial  lines  originally  straight  do  not  remain  straight  after  straining,  and  sections 

originally  plane  become  warped  under 

strain.      For  example,  Fig.   31   shows 

the  shape  assumed  by  each  section  of 

an  elliptical  shaft,  and  Fig.  32  indicates 

the   deformation   of    a   square    section 

under  strain.     The  net  result  of  this  is 

that  a  given  torque  produces  a  greater 

angular  displacement  than  is  indicated 

by  formula  (28),  and  the  angle  of  twist 

is  given  by 


Tl        Tl 


CkJ      CJ' 


.  (28A) 


where  J'  is  the  effective  polar  moment 
of  inertia  of  the  section. 

In  a  few  simple  cases,  where  the 
profiles  of  the  section  are  the  graphical 
representations  of  definite  mathematical 
functions,  values  of  J'  may  be  deduced 
from  considerations  of  strain,  and  Table  III.  shows  such  values  as  deduced  by  St.  Venant.1 

1  Todhnnter  and  Pearson,  "History  of  Elasticity,"  Vol.  II. 


Fro.  32. 


THE   TOESIONAL   EIGIDITY   OF   NOX-CIECULAE  SECTIONS      51 

TABLE  III. 


Type  of  Section. 


Remarks. 


Effective  value  of  «/(=  «/'). 


Solid  ellipse 


Major  axis  —  2a 
Minor  axis  —  2& 


Hollow  ellipse 


Major  axes,  2a  and  2aj 
Minor  axes,  26  and  '2b1 


Square 


Side  =  s 


•14s4 


Eectangle  . 


Lengths  of  sides,  b  and  d 


Any  symmetrical  section, 
including  rectangles,  in 
which  the  ratio  of  outside 
dimensions  in  any  two 
directions  in  a  cross- 
section  is  not  very  great 


A  —  area  of  section 
./  =  geometrical  polar 
moment  of  inertia 


It  becomes  apparent  from  St.  Venant's  investigation  that  there  is  always  greatest 
distortion  at  that  part  of  the  section  of  a  shaft  or  beam  under  torque,  where  the  surface 
is  nearest  the  axis.  The  distortion,  and  hence  the  intensity  of  stress,  becomes  very 
great  at  the  apex  of  any  re-entrant  angle,  becoming  infinite  where  the  apex  of  this  angle 


in 


FIG.  33. 


coincides  with  the  centroid  of  the  section.  On  the  other  hand,  the  distortion  and  stress 
in  the  neighborhood  of  projecting  points  is  very  small,  so  that  while  such  projecting 
areas  at  a  distance  from  the  axis  add  largely  to  the  magnitude  of  the  polar  moment  of 
inertia,  their  effect  on  the  tortional  resistance  of  the  section  is  usually  inconsiderable. 
Thus  such  sections  as  are  usual  in  I,  or  channel  beams,  and  which  offer  a  very  efficient 
distribution  of  material  to  resist  simple  flexure,  are  relatively  inefficient  to  resist 
torsion,  and  their  inefficiency  becomes  more  pronounced  as  the  distance  of  their  main 
members  from  the  centroid  of  the  section  is  increased. 

As  having  an  interesting  bearing  on  these  points  the  results  of  investigations  on 
the  following  sections  may  be  cited.     These  are  (Fig.  33) 

is  2 


52  A   STUDY   OF   THE   CIRCULAR- AEC   BOW-GIRDER 


(1)  Square  section. 

(2)  Ditto  with  slightly  concave  sides,  and  round  corners. 

(3)  Ditto          ditto  ditto          and  acute  corners. 

(4)  Star-shaped  section  with  four  rounded  points. 

T 'L 

Writing  6  =  — „  where  J',  the  effective  moment  of  inertia  of  the  section,  equals  kj, 

St.  Tenant  showed  that  the  values  of  k  for  these  sections  were  : — 


Section 

i 

2 

3 

4 

k 

•843 

•819 

•778 

•537 

FIG.  34. 


The  concavity  in  section  3  was  about  fa  of  the  length  of  the  side,  and  this  small  degree 
of  concavity  reduces  the  value  of  k  by  approximately  8  per  cent.  As  shown  by  the  value 
of  k  for  section  2,  this  concavity  has  more  influence  in  diminishing  the  torsional 
stiffness  of  a  beam,  for  the  same  moment  of  inertia,  than  the  rounding  of  the  corners  has 
in  increasing  it.  The  large  effect  of  a  greater  degree  of  concavity,  accompanied  by  the 

massing  of  material  in 
projecting  points  of  the 
section,  is  well  marked 
in  section  4.  As  com- 
pared with  a  circular 
section  of  the  same 
cross-sectional  area  and 
weight,  these  sections, 
offer  only  '891,  '867, 
•828  and  '674  times 
respectively  the  resist- 
ance to  torsion,  notwithstanding  the  fact  that  the  moments  of  inertia  of  their  section 
are  respectively  1*05,  1*06,  1P07,  and  T25  times  that  of  the  circular  section. 

St.  Tenant's  investigation  of  the  form  of  section  shown  in  Fig.  34  is  also  of 
interest.  This  section  consists  of  two  isolated  masses  of  material  symmetrically 
situated  with  respect  to  the  axis  of  twist ;  and  on  the  assumption  that  this  represents, 
the  section  of  a  beam  subjected  to  torque,  the  investigation  shows  that  the  value  of  k  is 
only  '0185.  This  section  approximates  more  or  less  closely  to  the  case  of  an  I  beam  in 
which  the  material  is  mainly  concentrated  in  the  flanges,  the  thickness  of  the  web 
being  small.  Comparison  between  this  value  for  k,  and  the  values  obtained  by 
experiment  on  I  sections  (see  Table  V.),  is  instructive.  It  is  evident  that  a  structural 
member  consisting  of  two  flat  bars  connected  by  a  lattice  bracing  must  of  necessity  be 
excessively  weak  in  torsion. 

For  complex  sections,  and  indeed  for  the  great  majority  of  commercial  sections, 
the  difficulties  involved  in  a  mathematical  investigation  of  the  value  of  J'  are 
insuperable,  and  such  values  can  only  be  determined  from  torsion  experiments. 

(29)  Experimental  Investigation  of  Torsional  Rigidity  of  Commercial  Sections. 

Such  experiments  have  been  carried  out  by  one  of  the  authors  and  are  described 
in  the  following  pages.  In  all,  twenty-one  beam  sections  were  tested.  The  details  and 


EXPERIMENTAL  INVESTIGATION  OF  TOKSIONAL  RIGIDITY      53 


dimensions  of  these  are  given  in  Table  I\r.  With  the  exception  of  the  solid  circular 
and  rectangular  sections,  and  the  welded  tubes,  which  were  of  wrought  iron,  all  were  of 
mild  steel. 

TABLE  IV. 


No. 

Section. 

Dimensions. 

Moments  of  Inertia  (ins.  units). 

Width. 

Depth. 

Thickness 
of  Flange. 

Thickness 
of  Web. 

Area. 
sq.  ins. 

,, 

,, 

J. 

1 

I 

5-01" 

8-02" 

•605" 

•30" 

8-02 

91  10 

13-10 

104-20 

2 

do. 

3-01" 

3-00" 

•325" 

•200" 

2-43 

3-70 

1-20 

4-90 

3 

do. 

1-75" 

4-78" 

•324" 

•190" 

1-91 

6-70 

0-26 

6-96 

4 

do. 

1-66" 

3-16" 

•23" 

17" 

1-222 

1-92 

•177 

2-097 

5 

do. 

•99" 

1-95" 

•165" 

•22" 

•6825 

•336 

•0281 

•3641 

6 

do. 

•76" 

1-50" 

•165" 

•14" 

•4141 

•1328 

•0124 

•1452 

7 

Channel 

•97" 

2-00" 

•227" 

•22" 

•7825 

•413 

•0618 

•4748 

8 

Angle 

1-175" 

1-175" 

•250" 

— 

•5245 

•0615 

•0615 

•1230 

9 

do. 

1-00" 

1-00" 

•185" 

— 

•3363 

•0275 

•0275 

•0550 

10 

Tee 

1-58" 

1-58" 

•231" 

•21" 

•650 

•1450 

•0739 

•2189 

11 

do. 

•99" 

•99" 

•135" 

•145" 

•2573 

•0236 

•0108 

•0344 

Solid 

12 

Rectangular 

•87" 

1-96" 

— 

— 

1-70 

•5460 

•1075 

•6535 

13 

do. 

•51" 

1-62" 

— 

— 

•827 

•1810 

•0180 

•1990 

Solid 

14 

Square 

•96" 

•96" 

— 

— 

•920 

•0702 

•0702 

•1404 

Hollow 

15 

Rectangular 

•872" 

1-432" 

X  -0360"  thick 

•151 

•0479 

•0223 

•0702 

Hollow 

16 

Square 

1-500" 

1-500" 

X  -0502"     „ 

•296 

•1035 

•1035 

•2070 

Solid 

17 

Circular 

1-01"  dia. 

•801 

•0510 

•0510 

•1020 

18 

do. 

•876"  dia. 

•601 

•0288 

•0288 

•0576 

Hollow 

Circular 

19 

(Welded) 

O.S.  dia.  1-305"  I.S.  dia.  1-05" 

•473 

•0826 

•0826 

•1652 

Hollow 

Circular 

20 

(Solid-drawn) 

O.S.  dia.  1-005"  I.S.  dia.    "923" 

•124 

•0144 

•0144 

•0288 

Hollow 

Oval 

21 

(Solid-  drawn) 

•862"  X  1-74"  X  -045"  thick 

•1788 

•0515 

•0173 

•0688 

The  method  of  carrying  out  the  torsion  tests  was  as  follows. — The  beam  under 
test  was  mounted  between  the  centres  of  a  six-foot  lathe,  centre-pops  being  made  on 
the  ends  of  the  beam  at  the  centre  of  gravity  of  the  section,  to  receive  the  lathe 


54  A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 

centres.  To  one  end  of  the  beam  was  clamped  a  lever  from  which  was  suspended  a 
hanger  fitted  with  a  knife  edge,  and  carrying  the  load.  Two  pointers,  each  three  feet 
long,  could  be  clamped  to  the  beam  at  any  desired  position.  These  pointers  moved 
over  scales,  clamped  to  the  bed-plate,  and  graduated  in  degrees  and  minutes.  Readings 
were  taken  to  the  nearest  minute.  The  other  end  of  the  beam  was  clamped  to  the 
head  of  the  lathe,  the  gear  being  locked  to  prevent  rotation. 

On  the  addition  of  each  increment  of  load,  scale  readings  were  taken  at  both 
pointers.  In  order  to  eliminate  the  effect  of  friction  at  the  centres,  the  torque  lever 
was  elevated  slightly,  and  allowed  to  decend  slowly,  depressed  slightly,  and  allowed  to 
rise  slowly,  the  angle  of  mean  position  being  noted.  Observations  were  made  for  both 
loading  and  unloading,  and  the  mean  angle  of  twist  per  unit  of  load  so  obtained.  The 
value  of  the  product  of  C  and  J'  was  then  found  from  the  formula. 

CJ'  -  Tl 

•~e 

where  the  symbols  have  the  significance  already  ascribed  to  them. 

In  each  case  the  experiment  was  repeated  over  a  span  of  about  half  the  original 
span.  In  no  case  did  the  two  values  of  CJ'  so  obtained  differ  by  more  than  3  per  cent. 

The  values  of  the  product  of  E  and  /  were  also  determined  by  supporting  the 
beam  on  two  massive  knife-edges  firmly  bolted  to  the  bed-plate;  Load  was  applied  to 
a  hanger  fitted  with  a  hardened  point,  suspended  from  the  middle  point  of  the  beam. 
Deflections  were  measured  by  means  of  a  micrometer  microscope  sighted  on  to  a  silk 
fibre  fixed  to  the  beam.  These  deflections  were  observed  to  the  nearest  *001  inch. 
Readings  were  taken  for  both  loading  and  unloading,  and  the  mean  deflection  per  unit 
load  calculated.  The  value  of  El  was  then  found  from  the  relationship 


In  order  to  obtain  the  values  of  the  two  moduli  E  and  C,  specimens  were  cut 
from  the  thickest  part  of  each  section,  turned  down  to  a  diameter  of  about  '18  inch, 
and  cut  to  a  length  of  about  9  inches.  The  values  of  C  were  then  found  by  means  of 
a  small  torsion  meter,  and  the  values  of  E  determined  by  supporting  the  specimens 
on  knife-edges  and  applying  a  load  at  the  middle  of  the  span.  The  values  of  the 
constants  so  found  have  been  tabulated  in  Table  V.,  which  also  shows  the  results  of  the 
torsion  and  bending  experiments  on  the  beams. 

The  Bending  Tests  show  that  in  general  the  experimental  and  theoretical  values  of 
E  I  agree  closely.  In  the  few  cases  where  a  fairly  large  discrepancy  exists  between 
them,  it  is  probably  due  mainly  to  the  fact  that  the  section  was  not  perfectly  uniform 
throughout  the  length  of  the  beam.  These  figures  indicate  roughly  the  discrepancy 
that  might  be  expected  from  calculations  based  on  the  ordinary  suppositions  that  a 
beam  is  of  uniform  section  throughout,  and  is  perfectly  straight  from  end  to  end. 

One  point  of  considerable  interest  is  brought  out  in  the  above  tests.  It  will  be 
observed  that  in  the  case  of  the  I,  channel,  and  other  sections,  the  values  of  E  obtained 
are  not  equal  for  both  axes  of  bending.  In  the  case  of  the  large  I  section,  for  instance, 
the  observed  values  of  E  when  the  web  is  vertical  and  when  the  web  is  horizontal  are 
respectively  30'7  X  106  and  26'4  X  106  in.-lb.  units.  In  the  former  case,  the  web  pro- 
vides 14'5  per  cent,  and  in  the  latter  case  only  '64  per  cent,  of  the  total  moment  of  inertia. 
Generally  speaking,  therefore,  the  modulus  of  elasticity  of  the  metal  in  the  flanges  is  less 
than  that  of  the  metal  in  the  web  ;  this  want  of  uniformity  being  undoubtedly  produced 
in  the  process  of  rolling.  This  is  confirmed  by  the  results  of  experiments  by  Prof.  E.  Mar- 


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EXPERIMENTAL  INVESTIGATION  OF  TOESIONAL  EIGIDITY      55 

burg,1  in  which  tension  test  pieces  were  cut  from  the  flange,  web,  and  root,  of  several 
I  beam  sections.  Tests  on  these  specimens  showed  a  considerable  variation  in  E  at 
different  points  in  a  section,  and  indicated  generally  a  lower  value  of  E  for  the 
flanges  than  for  the  web.  The  minimum  value  of  E  was  invariably  obtained  from  the 
test  piece  cut  from  the  junction  of  web  and  flange.  In  the  authors'  experiments  the 
channel  section  was  tested  with  the  web  both  in  tension  and  in  compression,  and  it  is 
interesting  to  note  that  the  flexural  strength  is  the  same  in  each  case.  In  the  angle 
sections  also,  the  flexural  rigidity  is  sensibly  the  same  whether  the  flange  is  in  tension 
or  compression. 

The  Torsion  Tests  afford  substantial  confirmation  of  St.  Tenant's  deductions  as  to 
the  inefficiency  of  material  in  the  neighbourhood  of  projecting  points  and  of  sharp 
corners  in  a  beam  section.  The  extreme  weakness  of  all  commercial  sections  is 
apparent  from  the  figures  given  in  column  12  of  Table  V.  The  inefficiency  of  I  and 
channel  sections  is  especially  remarkable,  while  tee  and  angle  sections  are  little  better. 

The  hollow  circular  section  is  the  most  efficient  of  all  for  withstanding  torsion.  It 
is,  however,  inefficient  when  exposed  to  bending,  and  is  for  many  reasons  ill  adapted 
for  girder  work.  Next  in  order  of  efficiency  comes  the  box  section.  So  long  as  the 
ratio  of  depth  to  breadth  is  moderate,  this  is  equally  well  adapted  for  resisting  either 
torsion  or  bending,  and  would  appear  to  afford  the  most  economical  distribution  of 
material  when  both  are  to  be  resisted. 

Solid  and  Hollow  Rectangular  Sections. — Reference  to  Table  V.  shows  that  k  is 
sensibly  the  same  for  a  hollow  as  for  a  solid  square  section,  having  a  value  '86  in  the 
latter  and  '87  in  the  former  case.  The  theoretical  value  of  k  deduced  from  St.  Tenant's 
formula  for  a  solid  square  section  is  '84  which  is  in  close  agreement  with  the  experimental 
value. 

The  agreement  between  calculated  and  experimental  results  in  the  case  of  the  solid 
rectangular  sections  is  equally  close.  Thus  for  section  12  (Table  V.),  depth  -J-  breadth 
=  2'25,  St.  Tenant's  formula  gives  k  ='47  against  the  measured  value  '46,  while  for 
section  18,  depth  -=-  breadth  =  3'18,  the  theoretical  and  measured  values  of  k  are  each 
equal  to  '29.  For  the  hollow  rectangular  section  No.  15  (depth  -=-  breadth  =  1*64),  the 
experimental  value  of  k  is  '69,  while  St.  Tenant's  value  for  a  solid  section  with  the 
same  ratio  of  breadth  to  depth  is  '68. 

It  thus  appears  that  the  value  of  k  for  a  hollow  rectangular  section  is  sensibly  the 
same  as  that  of  a  solid  section  of  the  same  overall  dimensions ;  depends  only  on  the 
ratio  of  breadth  to  depth  and  not  on  the  thickness  of  the  walls ;  and  that  the  value 
is  practically  identical  with  St.  Tenant's  theoretical  value  for  the  corresponding  solid 
rectangle. 

Tallies  of  /•:  for  such  sections,  having  different  values  of  the  ratio,  breadth  -i-  depth, 
are  given  in  Table  TL,  while  Table  TIL  shows  how  the  effective  value  of  J  varies  with 
this  ratio  in  such  sections  having  the  same  area  or  weight  per  foot  run.  It  will  be 
noted  that  while  both  k  and  J'  diminish  with  an  increase  in  the  ratio,  the  relative 
diminution  of  J'  is  not  nearly  so  great  as  that  of  k.  The  relative  diminution  of  J'  is 
approximately  the  same  for  hollow  as  for  solid  sections  with  the  same  overall  dimensions. 

Owing  to  the  inefficiency  of  the  material  in  the  corners  and  at  the  ends  of  the 
flanges  of  a  typical  commercial  box  section  (Fig.  35)  under  torsion,  the  value  of  J  or  of 
J'  for  such  a  section  should  be  computed  not  on  the  whole  area  but  on  the  portion 
included  by  the  rectangle  abed. 

1  Engineering  News,  Vol.  62,  1909,  p.  168. 


56  A   STUDY   OF   THE   CIECULAE-AEC   BOW-GIRDER 

TABLE  VI. 


Ratio. 
Greater  Side,  2c 

Value  of  k 
in 
TT 

Ratio. 
Greater  Side,  2c 

Value  of  k 
in 
TT 

Lesser  Side,  2b. 

CJ'  =  k  —  . 

e 

Lesser  Side,  2b. 

or-*f. 

1-0 

•841 

5-0 

•135 

1-5 

•721 

5-5 

•113 

2-0 

•550 

6-0 

•096 

2-5 

•413 

7-0 

•073 

3-0 

•316 

8-0 

•057 

3'5 

•247 

9-0 

•045 

4-0 

•198 

10-0 

•037 

4-5 

•161 

20-0 

•010 

TABLE  VII. — EFFECTIVE  VALUES  OF  J  FOR  RECTANGULAR  SECTIONS  HAVING  THE 

SAME  CROSS-SECTIONAL  AREA. 


Ratio  ?£. 
'2f> 

2c. 

2b. 

Theoret.  J. 

k. 

Effective  JorJ'. 

1 

1-0 

1-0 

•166 

•841 

•140 

2 

1-416 

•708 

•209 

•550 

•115 

4 

2-00 

•500 

•354 

•198 

•070 

6 

2-448 

•408 

•511 

•096 

•049 

10 

3-160 

•316 

•917 

•037 

•034 

EXPEEIMENTAL  INVESTIGATION  OF  TOKSIONAL  EIGIDITY       57 


d 


Since  an  increase  in  depth  renders  a  section  more  efficient  to  resist  bending,  the 
most  effective  value  of  this  ratio  when  both  torsion  and  bending  are  to  be  resisted, 
depends  on  the  relative  values  of  the  two  moments.  With  zero  bending  moment  the 
section  should  be  square.  With  zero  torque,  expe- 
rience shows  that  the  ratio  of  breadth  to  depth 
should  be  between  3'5  and  5'0  for  best  results. 
With  both  torsion  and  bending  the  most  economi- 
cal ratio  will  usually  lie  somewhere  between  2O 
and  3'5,  its  value  increasing  as  the  ratio  of  bending 
moment  to  twisting  moment  increases. 

/  Sections. — A  comparison  of  the  results  of  the 
torsion  tests  on  I  sections  Nos.  1  to  6,  Table  V., 
indicates  that  the  ratio  of  actual  to  calculated 
value  of  ./  .diminishes  with  an  increase  in  the  size 
of  the  section.  The  penultimate  column  in  Table 
VIII.  gives  the  values  of  k  for  these  sections.  The 
value  of  J'  in  inch  units  is  given  with  a  fair  degree 
of  accuracy  by  the  relationship 


=  &>  (r  T 

FIG.  35. 

where  A  is  the  area  of  the  section  in  square  inches. 

The  last  column  of  this  table  shows  values  of  A2-^-  60,  while  experimental  values  of  J' 
are  given  in  column  6. 


TABLE  VIII. 


Section 
Number 
Table  V. 

Approximate 
Dimensions. 

2fl 

2o" 

Area  "  A  " 

J 

J' 

it 

A* 
60 

1 

8"  X  5" 

1-60 

8-02 

104-0 

1-04 

•010 

1-07 

2 
3 

4f"  X  If" 
3"  X  3" 

2-73 
1-00 

1-90 
2-43 

6-96 
4-90 

•058 
•099 

•0083 
•0202 

•060 
•098 

4 
5 

3"  X  H" 
2"  X  1" 

1-90 
1-97 

1-22 

•682 

2-10 
•364 

•024 
•0094 

•0114 
•0260 

•025 
•0078 

6 

1  1"   v    3" 
1.)      A    4 

1-97 

•414 

•145 

•005 

•0344 

•0029 

From  these  figures  it  appears  that  for  sections  1  to  4  the  formula  gives  results 
which  are  accurate  within  about  3  per  cent.  These  are  all  commercial  sections.  The 
agreement  is  not  so  close  for  section  5,  and  is  unsatisfactory  for  section  6.  These 
tsvo  are  not  commercial  sections,  and  the  relative  thickness  of  web  and  of  flanges  is  much 
greater  than  in  commercial  sections,  especially  in  section  6,  in  which  the  discrepancy 
is  most  pronounced.  Probably  for  all  normal  commercial  I  sections  expression  (29) 
will  give  results  sufficiently  accurate  for  purposes  of  design. 

Angle,  Tee,  and  Channel  Sections. — An  examination  of  the  results  of  the  tests  on  the 
angle,  tee,  and  channel  sections  of  Table  V.,  shows  that  the  value  of  k  varies  widely 


58 


A  STUDY   OF   THE   CIRCULAR  ARC   BOW-GIRDER 


with  the  type  of  section.     The  value  of  J'  is  given  within  about  2  per  cent,  in  every 
case  by  the  relationship 


„ 
«/'  = 


m 


(30) 


where  m  varies  with  the  type  of  section.     Values  of  k  and  of  m  are  given  in  Table.  IX. 

TABLE  IX. 


Section. 

Mean  value  of  k. 

HI. 

Channel     . 

•025 

40 

Tee    . 

•06 

25 

Angle 

•09 

18 

Compound  Girder. — Experiments  were  also  carried  out  on  a  compound  girder  of  the 
type  shown  in  Fig.  36.  This  consists  of  two  8"  X  4"  commercial  I  sections,  distant 
10'3  inches  centre  to  centre,  and  tied  together  at  intervals  of  2'  6"  by  plates  across  the 

bottom  flanges.   The  value 

ffTl  f  ftl rf^H f^Ti       °f  J  for  this  combination 

~n  is  370  inch  units ;  the 
value  of  J'  is  2'05  inch 
units  ;  and  the  value  of  k 
is  -0055.  Calling  A  the 
total  area  of  both  sections, 


Axis     oF 


Twist . 


-no 

as  compared  with  the  value 
a  single  girder  of 


A* 
60  for 


*-—^  the  same  total  weight  per 

FIG.  36.  foot  run  as  the  combined 

girder. 

Tests  on  Hollow  Box  Sections  filled  in  with  Concrete. — Since  in  a  hollow  box  section 
torsion  is  accompanied  by  distortion  of  the  webs  and  flanges  (Fig.  46)  it  was  anticipated 
that  by  filling  the  interior  of  such  a  section  with  concrete  this  relative  distortion  might 
be  reduced  to  some  extent,  and  the  section  be  stiffened  in  consequence.  To  test  this 
point  the  hollow  sections  Nos.  15  and  16,  Table  IV.,  were  filled  with  cement  grout  and, 
after  setting  for  four  weeks,  were  again  tested  in  torsion.  The  effect  of  this  is,  however, 
not  great.  E.g.,  with  section  (15),  J'  without  filling  was  '0483,  and  with  filling  '0508, 
while  in  section  (16)  J'  was  increased  from  '1645  to  '1941  by  the  filling. 


CHAPTEE  IV 
MAGNITUDE  OF  SHEAK  STRESSES  IN  A  BEAM  UNDER  TORSION 

(30)   Beam   of  Circular  Section. 

IN  a  beam  of  circular  section  the  shear  produced  by  torsion  is  everywhere  circum- 
ferential, and  varies  directly  as  the  distance  from  the  axis  of  twist.  Thus  if  /  be  the 
magnitude  of  this  shear  at  a  radius  r,  and  fs  its  magnitude  at  the  surface  where  the 
radius  is  a,  we  have 

T 

f—    f         - 
J-Js    •    a- 

The  moment  of  the  shear  on  an  elementary  concentric  ring  of  radius  r  and  of  radial 
width  Br  will  therefore  be 

2m*  .  /.  .  Br 

a 

and  on  integrating  this  expression  over  the  whole  section  of  the  beam  and  equating  the 
result  to  the  external  torque  T,  we  have 

/.=  £ (3D 

Here  fs  is  the  maximum  circumferential  shear  in  the  section.  This  formula  is 
applicable  to  both  solid  and  hollow  circular  sections. 

(31)  Sections    other  than  Circular. 

In  a  non-circular  section  under  torsion  the  assumptions  that  the  shear  at  any 
point  is  perpendicular  to  the  radius  at  that  point  and  is  proportional  to  its  distance 
from  the  axis  of  twist,  are  no  longer  true.  It  has  been  shown  both  by  St.  Venant  and 
by  Bach  1  that  the  maximum  transverse  shear  stress  in  any  non-circular  section  under 
torque  occurs  at  that  point  on  the  surface  which  is  nearest  to  the  axis  of  twist ;  that 
the  stress  is  great  in  the  neighbourhood  of  re-entrant  angles  and  zero  in  the  neighbour- 
hood of  projecting  corners. 

Expressions  for  the  maximum  shear  in  the  case  of  a  few  of  the  simpler  sections 
such  as  the  ellipse  and  the  rectangle  have  been  deduced  by  St.  Venant,  and  are  given 
on  p.  72.  Autenreith 2  assumes  that  the  stress  at  a  given  point  P  (Fig.  37)  on  the 
boundary  of  any  solid  or  hollow  section  bounded  by  a  continuous  curve  convex 
outwards,  is  given  by 

9TT 

f.  =  jf  •     (82) 

where  T  is  the  torque,  A  the  area  of  the  section,  and  r  is  the  length  of  the  perpen- 
dicular from  the  centroid  of  the  section  on  to  the  tangent  at  P.  The  maximum  shear 
stress  will  thus  occur  where  r  is  a  minimum,  i.e.,  at  the  end  of  the  minor  axis  of  the 
section,  and  the  minimum  surface  shear  at  the  end  of  the  major  axis. 

On  the  same  assumptions  the  surface  shear  in  a  hollow  section  having  a  continuous 

1  "Elastizitat  und  Festizkeit." 

2  Zeitschrift  des  Vereines  deutscJter  Inyenienre,  1901,  p.  1099 


60 


A   STUDY   OF   THE   CIKCULAK-ABC   BOW-GIKDEK 


boundary,  in  which  the  ratio  of  inner  to  outer  radius  is  sensibly  constant  and  equal 
to  y  for  all  radii,  is  given  by 

27' 


/.= 


.     (33) 


(32)   Solid  and  Hollow  Elliptical  Sections. 

For  a  solid  or  hollow  elliptical  section,  having  semi-major  and  minor  axes  a  and  b, 
the  value  of  ;•  at  any  point  P  whose  co-ordinates  are  xy  (Fig.  37)  is  given  by 


.34) 


SOLID   AKD   HOLLOW   ELLIPTICAL   SECTIONS 


61 


In  a  hollow  section  having  a  and  b  as  the  semi-major  and  semi-minor  axes  of  its  external 
surface,  the  area  of  section  is 

,    .        ai       bi 

TT  [ab  —  a\bi\ ,  and  since  —  =  —  =  •/ 

a        o 

/.  A  =Trab{l  —  y2} 


/•/  1-0  -9  -8  -7  -6  -5  -4   -3   -2   •/ 


Note.-   Intercepts   of     Normals     Give     Values    of  y 


FIG.  38. — Diagram  showing  distribution  of  surface  shear  stress  in  a'solid  elliptical  section  subjected 

to  a  twisting  moment. 


Thus  in'the  general  case 


J * 


-nab  [I  -y2]  [1  +  y2]  r 
27Vv/2(a2  —  b2)  +  />4 


y4] 


•     (35) 


62  A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 

and  for  a  solid  elliptical  section  (y  =  0)  this  becomes 


_  arvv  -  ip)  + 

''*  ~  ird'b* 


•     (36) 


Note:-  Intercepts   of  Normals  Give    Values   of 


FIG.  39. — Diagram  showing  distribution  of  surface  shear  stress  in  a  hollow  elliptical  section 

subjected  to  a  twisting  moment. 

The  maximum  shear  occurs  at  the  end  of  the  minor  axis  where  y  =  b,  and  is 
given  by 


277 


/(max.)  — 


which  agrees  with  St.  Venant's  result. 

The  minimum  stress  on  the  periphery  is  given  by 


•     (37) 


22' 


lniuj 


y4) 


SOLID   AXD   HOLLOW   ELLIPTICAL   SECTIONS 


63 


Where  a  =  b  =  r,  each  of  these  expressions  reduces  to 

27V 


f 


77  \1 


.4  _ 


.     (38) 


the  expression  for  the  shear  at  the  periphery  of  a  hollow  circular  section. 

Figs.  38  and  39  show  respectively  the  distribution  of  surface  shear  in  a  solid  and 
a  hollow  elliptical  section,  in  each 
of  which  a  :  b  =  1'5,  while  y  =  '934. 
These  are  subject  to  the  same  torque 
and  have  the  same  cross  sectional 
area.  The  magnitude  of  the  stress 
is  indicated  by  the  normal  to  the 
surface,  intercepted  between  the  sur- 
face and  the  curve.  In  this  case  the 
maximum  stress  in  the  solid  section 
is  5  times  as  great  as  in  the  hollow 
section. 

In  a  solid  circular  section  of  the 
same  area  the  maximum  stress  is  "82 
times  that  in  the  solid  elliptical  sec- 
tion, while  in  a  hollow  circular  section 
having  the  same  thickness  and  the 
same  area  as  the  hollow  elliptical 
section,  the  maximum  stress  is  "76 
times  that  in  the  latter  section. 

While  the  assumptions  made  in 
deducing  the  foregoing  formulae  give 
results  in  close  agreement  with  ex- 
periment if  the  boundary  is  a  con- 
tinuous curved  line,  they  fail  to  do  so 
if  the  section  has  a  discontinuous 
boundary.  In  the  latter  case  the  re- 
searches of  Bach  indicate  a  state  of 
zero  stress  at  projecting  points,  and, 
in  an  extreme  case  would  postulate 
zero  stress  at  the  corners  of  a  poly- 
gonal section  no  matter  how  closely 
this  approximates  to  a  circle.  To 
obviate  this  difficulty  Autenreith  as- 
sumes that  the  stress  at  such  a  corner  depends  upon  the  included  angle,  being  zero 
for  a  right  angle,  and  that,  at  any  point  in  the  surface  of  such  a  section  in  which  this 
angle  is  not  less  than  90°,  it  is  given  by 


FIG.  40. 


-  2#  fi        (* 
r    L 


sm  a 


.     (39) 


where  /;  is  the  circumferential  shear  stress ;  r  the  length  of  the  perpendicular  from 
the  centroid  to  the  corresponding  side  of  the  polygon  ;  0  a  constant ;  z  the  distance 
from  the  mid  point  of  the  side  to  the  point  at  which  the  stress  is  required  ;  c  half  the 
length  of  the  side ;  and  a  is  the  included  angle  (Fig.  40). 


64  A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 

When  a  =  180°,  i.e.,  for  a  circular  section,  this  makes  /,  =  constant.  When 
a  =  90,  i.e.,  for  a  square  or  rectangular  section,  fs  becomes  zero  when  z  =  c  (at 
corner),  and  attains  a  maximum  value  when  z  =  0,  i.e.,  at  the  centre  of  the  side.  In 
these  two  extreme  cases  the  formula  thus  agrees  with  the  results  of  experiment. 
Assuming  that  at  any  point  in  the  interior  of  the  section  the  component  of  the  shear 
stress  normal  to  the  radius  vector  is  proportional  to  the  distance  from  the  centroid,  an 
expression  may  be  obtained  for  the  moment  of  the  shear  on  any  element,  and  on 
integrating  this  over  the  whole  section  and  equating  to  the  torque  the  value  of  the 
constant  ft  may  be  obtained. 

This  is  given  by 

.....     (40> 


. 


where  A  is  the  area  of  the  section. 

Since  p  sin  <£  =  z  (Fig.  40)  equation  39  becomes 

867'  L        /psin</>2 


For  a  hollow  polygonal  section  in  which  the  ratio  of  inner  and  outer  radii  vectores 
is  sensibly  constant  and  equal  to  7,  this  formula  becomes 

36 T |    _  fp  sin  <ft\  2   .      ) 

*  "~  rA  [18  (1  +  7*)  —  4  sm  a  (I  +  ^  +  y*;]  (          \      c      ) 

In  each  case  the  maximum  shear  occurs  at  the  middle  of  the  side  of  the  polygon  where 

</>  =  0,  and  is  given  by  -— ,  where,  for  a  solid  section, 
AT 

li  = ^-. (43) 

9  —  2  sm  a 

and,  for  a  hollow  section, 

ift 

•     (44) 


9(1  +  7*)  -  2  sin  a  (1  +  y*  +  y4) 

(33)  Rectangular    Sections — Box   Sections. 

In  a  solid  rectangular  section  (Fig.  41),  whose  longer  side  is  2  e  and  shorter  side 
2  b,  r  for  the  shorter  side  is  c,  and  for  the  longer  side  is  1>.  Also  sin  a  =  l,  so  that, 
for  the  longer  side  equation  41  becomes 


and  for  the  shorter  side 


Thus  the  maximum  stress  in  the  longer  side  (at  its  mid  point,  where  </>  =  0)  is  given  by 

/•_,:=  2-57-^  .     (46) 


and  the  maximum  stress  in  the  shorter  side  by 
At  the  corners  in  each  case/,  =  0. 


f        —  2-57  T.  .  (46A) 

./(max.)  —  «  «" 


BECTANGULAK   SECTIONS— BOX  SECTIONS 


65 


In  the  case  of  a  hollow  rectangular  or  box  section  in  which  y  is  sensibly  constant 
equation  (42)  applies.     The  shear  at  any  point  in  the  longer  side  is  given  by 

18  T  L       /psin 


from  which 


Ab[l(l  4-  y2)  -  2y4]  v 
1ST 


/(max.)  — 


•     (47) 
(48) 


FIG.  41. 


while  for  the  shorter  side 

'.-= 

and 


1ST 


Ac[l(l  +  i 

/(max.)  — 


sn 


1ST 


+  y2)  -  2y4]   ' 


.     (49) 
.     (50) 


From  equations  (45)  and  (47)  it  appears  that  the  curves  of  stress  distribution  in  a 
rectangular  section  are  parabolic. 

Figs.  42  and  43  show  such  curves  drawn  respectively  for  a  solid  and  a  hollow 
rectangular  section  having  the  same  ratio  1'5,  of  depth  to  breadth,  and  the  same  cross 
sectional  area  and  weight  per  foot  run.     In  the  hollow  section  the  ratio  of  inside  to 
E.G.  F 


66 


A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


outside  dimensions,  or  y,  is  '975.     From  these  curves  it  appears  that  the  maximum 
stress  in  the  box  section  is  about  19  per  cent,  of  that  in  the  solid  section. 

Comparing  diagrams  39  and  43,  it  appears  that  the  ellipitical  section  is  the  more 
efficient  in  that  the  maximum  stress  is  only  72%  of  that  in  the  box  section.      In  the 


FIG.  42.  —  Diagram  showing  variation  in  surface  shear  stress  in  a  solid  rectangular  section  sub- 
jected to  a  twisting  moment. 

Ratio 


breadth 


=.1-5  ;  Area  of  section  =  2-4. 


ordinary  box  section  used  in  practice  the  value  of  y  will  not  in  general  be  the  same  for 
the  top  and  bottom  flanges  as  for  the  webs,  nor  can  it  be  the  same  for  different  points 
on  web  or  flange  since  these  are  of  uniform  thickness.  From  the  following  table, 
which  shows  calculated  values  of  il  in  the  formula 


./(max.)  — 


(51) 


EECTANGULAB   SECTIONS— BOX   SECTIONS 


67 


for  a  hollow  box  section  4  ft.  square  and  with  different  thicknesses  of  metal,  it  appears, 
however,  that  a  given  percentage  variation  in  y  only  produces  about  one-half  the  same 


st-ss  ia  a  rectangular  box-section  sub- 


Eatio 


breadth 

7=  '975. 
Area  of  section  =  2-4. 


percentage  variation  in  O.     In  practice  the  mean  of  the  values  of  7  measured  at  the 
mid  points  of  the  two  sides  will  give  results  within  a  few  per  cent,  of  the  truth. 


p  2 


68  A   STUDY   OF   THE   CIECULAE-AEC    BOW-GIEDEE 


Thickness 
of  Metal. 

i" 

i" 

1" 

i" 

H" 

H" 

7 

•989 

•978 

•968 

•958 

•947 

•937 

A 

47-75 

95-0 

141-7 

188-0 

233-7 

279-0 

O 

1-510 

1-517 

1-532 

1-538 

1-542 

1-548 

The  foregoing  investigations  of  Autenreith  are  based  upon  a  consideration  of  the 
stresses  involved  during  torsion.  St.  Venant,  considering  the  strains  produced, 
obtained  the  expression 


rise  +  9/n 

/(max.)-        _     40c.2/>2     _ 


_OT 

=  Ab 

for  the  maximum  shear  stress  in  a  rectangular  section  of  sides  2c  and  2/>.      In  this 
formula 

12  =  1-5  +  0-9  -. 
c 

Table  IX.  shows  how  11  varies  with  the  ratio  of  depth  to  breadth. 

TABLE  IX. 


Ratio  -. 
o 

i 

2 

3 

4 

6 

<; 

7 

8 

9 

KI 

li  (St.  Venant)  . 

2-40 

1-95 

1-80 

1-72 

1-68 

1-65 

1-63 

1-62 

1-60 

1-59 

According  to  Autenreith  il  is  independent  of  the  ratio  e-i-  />,  and  has  a  constant 
value  2'57,  so  that  stresses  calculated  from  Autenreith's  formula  are  greater  than 
those  obtained  by  St.  Venant,  the  difference  becoming  more  pronounced  as  this  ratio 
is  increased. 

Bach's  experiments  on  the  whole  appear  to  show  that  Autenreith's  values  are  in 
closer  accord  with  the  result  of  experiment,  and  for  purposes  of  design  these  may  be 
adopted  with  some  confidence.  The  calculated  stresses,  if  they  err  at  all,  will  do  so  on 
the  side  of  safety. 

(34)  I    Sections. 

Little  definite  is  known  as  to  the  magnitude  and  distribution  of  stress  in  a 
member  of  I  section  under  torque,  except  that  the  stress  is  greatest  at  the  mid  point 


I   SECTIONS  69 

of  the  web  and  is  zero  at  the  extremities  of  the  flanges.  Since  the  stress  is  always 
large  in  the  neighbourhood  of  a  re-entrant  angle,  it  is  probable  that  it  will  be  large  at 
the  junction  of  web  and  flange,  particularly  where  the  radius  of  the  fillet  at  this  point 
is  small.  As  to  this  point,  however,  no  definite  information  is  available. 

From  experiments  on  I  sections  made  of  lead  Bach  found  that  rupture  always 
occurred  at  that  point  on  the  web  nearest  to  the  centroid  of  the  section,  and  deduced 
the  expression 

/^ax,  =  4-5^ (52) 

where  A  is  the  total  area  of  the  section  and  t  is  the  thickness  of  the  web. 

Some  confirmation  of  this  formula  has  been  obtained  by  the  authors.  Thus 
considering  I  section  No.  1  (Table  IV.),  the  effective  value  of  J'  for  the  whole  section  is 
T04,  while  J'  for  the  web  if  isolated  from  the  rest  of  the  section  would  be  approximately 
•086.  Adopting  these  values,  the  web  may  be  expected  to  take  approximately 

— —  =  '082  of  the  total  torque,  and  from  formula  (46),  p.  64,  the  maximum  stress  in 

the  web  would  then  be  equal  to 

2-57  X  '082  T 
Ab 

where  b  is  the  half  thickness  of  the  web,  or  ^. 

a 

On  making  this  substitution  the  formula  becomes 

4-2  T 


/(max.)  — 


At 


which  is  in  fair  agreement  with  Bach's  expression  for  the  same  stress. 

Although  the  stress  at  other  parts  of  the  section  is  indeterminate,  experiment 
shows  that  if  the  web  is  made  stiff  enough  to  withstand  this  stress  the  remainder  of 
the  section  is  amply  strong. 

(35)  Horizontal   Shear   in   a  Beam   Subject   to   Torsion. 

What  aver  be  the  magnitude  of  the  transverse  shear  stress  due  to  torsion  at  a 
point  in  a  vertical  section  of  a  horizontal  beam,  this  shear  will  be  accompanied  by  an 
equal  shear  stress  on  the  horizontal  plane  passing  through  the  same  point.  In  a 
beam  of  box  section  in  which  the  depth  exceeds  the  breadth,  or  in  a  beam  of  I  section, 
the  magnitude  of  this  shear  on  horizontal  layers  is  a  maximum  at  the  neutral  axis. 

(36)  Resultant   Shear   on   Horizontal   and   Vertical   Sections   of   a   Beam    Exposed 

to   Torsion   or   Bending. 

The  resultant  shear  at  any  point  in  a  horizontal  or  vertical  section  of  a  beam  is 
the  algebraic  sum  of  the  shears  due  respectively  to  bending  and  to  torsion.  The  shear 
stress  due  to  torsion  has  already  been  discussed.  The  shear  stress  due  to  bending,  or 
to  the  application  of  the  vertical  loads  and  reactions  which  produce  bending,  varies 
from  point  to  point  in  a  section. 

If  q  denotes  the  intensity  of  shear  due  to  this  vertical  loading  at  a  point  distant 


70 


A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


~~!  from  the  axis  of  bending,  and  if  the  breadth  of  the  section  at  this  point  be  ylt  this 
shear  stress  is  given  by  l 


•     (38) 


where  F  is  the  shear  force  at  the  section  in  question,  and  z2  is  the  distance  of  the 
outer  fibres  of  the  section  from  the  neutral  axis. 


FIG.  44. 

In  a  rectangular  section  of  breadth  26  and  depth  2c, 
while  if/2  =  c,  and  expression  (53)  becomes 

Tc 
2fd^ 


=  //i  =  2/>  is  constant, 


2i 

J 


(54) 


This  distribution  of  shear  over  the  section  is  parabolic.     The  maximum  value  occurs 

q    77*         q   T,T 

at  the  neutral  axis  where  z\  =  0,  and  is  equal  to  -    -  or  -  — ,  or  to  1'5  times  the  mean 

o  be        '—  A 

shear  over  the  section.     The  minimum  value,  zero,  occurs  at  the  outer  extremity  of 
the  section  where  z1  =  c. 

(37)  I    and    Box   Sections. 

In  the  case  of  an  I  or  rectangular  box  section  the  breadth  is  constant  over  the 
web  and  is  suddenly  increased  at  the  flanges.  As  a  result  of  this  the  magnitude  of 
the  shear  stress  in  the  flanges  is  much  less  than  that  in  the  web.  The  distribution  of 
this  stress  is  indicated  in  Fig.  44.  In  an  average  section  the  intensity  of  stress  in  the 

1  Moiiey,  "  Strength  of  Materials,"  Chapter  V. 


I  AXD   BOX  SECTIONS 


71 


web  does  not  change  greatly,  and  the  usual  assumption  that  the  web  carries  the  whole 
vertical  shear  force  with  uniform  distribution  gives  stresses  which  are  in  fair  agreement 
with,  and  usually  slightly  higher  than  those  actually  attained. 

In  a  hollow  box  section  formed  by  the  rectangles  2&,  2c,  and  2&i,  2ci,  or  in  the 
corresponding  I  girder  (Fig.  44),  in  the  flange  at  a  height  z1  from  the  neutral  axis. 


3          F 


8  [be3  —  blCl3] 


.     (55) 


while  in  the  web  at  a  height  zlt 

3           F  \b(c2  — 

" 


q  ~  8  [be3  —  b^3}  !      b  —  hi 


.21 


and,  at  the  neutral  axis, 


(/(max.) 


3  F 


$  [be3  —  blCl3]       b  - 


.         .     (56) 

(57) 


It  should  be  noted  that  whereas  the  shear  on  a  vertical  section  produced  by  the 
vertical  loading  acts  in  the  same  direction  at  all  points  in  the  section,  that  due  to 
torsion  acts  in  opposite  directions  at  opposite  ends  of  a  diameter.  It  follows  that  the 
shear  stresses  due  to  bending  and  torsion  act  in  the  same  direction  in  one  of  the  webs 
of  a  box  girder,  and  in  opposite  directions  in  the  other,  and  that  under  such  combined 
moments  one  web  will  be  much  more  heavily  stressed  than  the  other. 

The  nature  of  the  resultant  shear  stress  distribution  over  the  vertical  section  of 
such  a  girder  is  indicated  by  the  curves  of  Fig.  45. 


72  A   STUDY   OF   THE   CIRCULAR-ARC  BOW-GIRDER 

TABLE  X. 


Type  of  Section. 

Maximum  surface  shear  stress. 

St.  Venant. 

Autenrieth. 

Solid  Circular 
Eadius  r 

2T 

TTT3 

277 

Try3 

Hollow  Circular 
radii 
TI  and  r2 

2Tyi 

2ryi 

^[n4  -  ra*] 

wtn4  -  ^4] 

Solid  Elliptical 

T\TrHor  Avifi  —  9,/* 

2T 

2T 

Minor     „    =  26 

7TC62 

7TC62 

Hollow  Elliptical 
formed  by 

[2c  26]  [2c0  260] 

— 

2T6 

77  [c63  —  c0603] 

Solid  Eectangular 
Long  Side  =  2c 
Short  Side  =  26 

T15c  +  961  T 

•643^ 

t-62 

L  40c262  J 

Hollow  Kectangular 

is              r 

Short  side  =  26 

[7|l+72j  -VI  '  ^6 

Any  Polygonal  Section 
Had.  of  InsfHbpd  CirnlA  —  »• 

— 

18                                T 

Included  Angle  =  a 

[9{l  +  y2}  —  2sina[l  +  y2  +  y4]]  '  Ar 

I 
Web  Thickness  =  t 

— 

«z 

CHAPTER   V 


(38)  General   Principles    of    Design    of  the    Bow-Girder, 

FROM  the  data  of  Chapters  III.  and  IV.,  it  appears  that  where  a  beam  is  exposed 
to  any  appreciable  torsion,  the  box  section  is  from  every  point  of  view  the  most  suitable, 
and,  for  beams  of  considerable  span,  or  carrying  heavy  loads,  is  the  only  practic- 
able section.  For  comparatively  small  spans  ;  for  spans  in  which  the  radius  of 
curvature  is  large  and  the  angle  sub-tended  by  the  arc  between  successive  supports  is 
small,  or  for  moderate  loads,  the  I  section  may  be  permissible,  but  in  general  its  use 
is  to  be  deprecated  wherever  combined  torsion  and  bending  is  anticipated. 

In  any  case,  where  not  barred  by  other  considerations,  intermediate  supports  are, 
as  shown  by  the  results  of  the  investigations  in  Chapter  II.,  of  the  greatest  value  in 
reducing  the  applied  moments,  and  especially  the  twisting  moment  at  a  given  section. 

In  a  box  section  exposed  to  twisting  and  bending,  a  general  consideration  of  the 
problem  indicates  that  most  economical  results  are  to  be  obtained  where  the  ratio  of 
depth  to  breadth  has  a  value  somewhere  between  2'0  and  3'5,  the  former  value  applying 
to  encastre  beams  without  intermediate  supports  and  subtending  an  angle  in  the 
neighbourhood  of  180°,  and  the  latter  for  beams  adequately  supported  at  intermediate 
points  or  subtending  angles  not  exceeding  45°.  The  following  may  be  taken  as 
affording  a  first  approximation  to  the  relative  dimensions  of  such  a  girder  designed  for 
heavy  duty : — 


Angle  subtended  by  arc  between  supports. 

180° 

150° 

120° 

90° 

60° 

30° 

depth 

2-0 

2-25 

2-5 

2-75 

3-0 

3-25 

breadth 

Having  assumed  a  suitable  section  for  the  girder,  the  tensile  and  compressive  stresses 
due  to  the  bending  moment,  and  the  shear  stresses  due  to  the  vertical  loading,  are 
to  be  determined  for  each  section  of  the  girder,  as  in  the  case  of  a  straight  girder,  the 
value  of  the  bending  moment  being  obtained  from  the  data  of  Chapter  II.  The  value 
of  the  twisting  moment  at  each  section  having  been  calculated  in  the  same  way,  the 
shear  stress  due  to  this  may  be  determined  by  an  application  of  the  results  of 
Chapter  III.,  and  this  shear  stress  is  to  be  added  to  the  shear  stress  due  to  the  vertical 
loading,  to  give  the  actual  shear  at  a  given  point  in  the  section.  In  the  box  or  I  section 
both  components  of  shear  have  their  maximum  value  at  the  neutral  axis.  The  shear 
in  the  flanges  of  such  a  girder,  due  to  the  vertical  loading,  is  sensibly  zero.  That  due 
to  torsion  is  in  general  also  small,  and  where  the  flanges  are  of  adequate  thickness  to 
withstand  the  direct  stresses  due  to  bending  there  is  little  question  as  to  their  ability  to 


74  A   STUDY   OF   THE   CIRCULAR-ARC   BOW-GIRDER 


take  care  of  the  additional  small  stress  due  to  torsion.  Having  obtained  the  resultant 
shear  in  the  webs,  these  should  be  designed  by  the  ordinary  rule  applicable  to  the  web 
of  a  straight  plate-web  girder  subject  to  the  same  stress.1 

Under  torsion  such  a  girder  tends  to  buckle  as  shown  by  the  dotted  lines  of 
Fig.  40,  and  particular  attention  should  be  paid  to  stiffening  the  webs  against  this 
action.  Under  normal  circumstances  this  may  be  accomplished  by  the  use  of  angle  or 
tee  stiffeners,  between  flanges,  reinforced  if  necessary,  where  the  torsion  is  greatest, 
by  the  addition  of  a  cover-plate  to  the  web. 

The  pitch  of  the  stiffeners  should,  strictly  speaking,  diminish  as  the  torsion 
increases.  Where  torsion  is  large  the  pitch  should  not  exceed  the  depth  of  the  girder, 
for  girders  less  than  2  feet  6  inches  deep,  and  should  not  exceed  about  one  half  the 
depth  for  a  girder  6  feet  deep. 

Special  attention  should  be  paid  to  the  design  of  the  riveting  at  the  junction  of 

web  and  flange,  since  this  has  not  only  to  with- 
stand a  shear  of  magnitude  equal  to  that  of  the 
vertical  shear  at  this  point,  but  has  also  to  resist 
the  tendency  to  relative  distortion  indicated  in 
Fig.  46.  This  latter  effect  also  involves  the  use 
of  somewhat  heavier  angle  sections  than  are  usual 
in  the  straight  girder. 

Where  joints  in  the  web  plates  are  necessary 
these  should  be  placed  where  the  sum  of  tor- 
sional  and  load  shear  is  a  minimum. 

As  an  example  the  preliminary  design  of  a 
bow  girder  of  uniform  section  of  30  feet  radius, 
built  in  at  the  ends  and  subtending  an  angle  of 
120°,  and  carrying  a  uniform  load  of  2  tons  per 
foot  run,  may  be  considered.  The  values  of  Me 
and  TO  for  such  a  girder  having  El  :  CJ  =  T25, 
,  are  given  by  the  curves  of  Figs.  22  and  23,  $ 
being  30°.  From  these  curves  it  appears  that 
MO  has  its  maximum  value  (•  42  /rr2)  at  the  support, 

while  at  this  point  T  =  '048  irr2.  The  maximum  value  of  Tg  ('052  wr2)  occurs  at 
approximately  30°  from  the  support,  but  since  at  this  point  MO  is  zero,  and  since  the 


at 


support, 


vertical  shear  force  is  only  wr  I5~~^"~gj  as  against  wr     5  —  <t> 

the  latter  will  be  the  point  of  maximum  resultant  stress. 

Preliminary  investigation  indicates  that  a  box  girder  5  feet  deep  and  2  feet  wide, 
with  flanges  \\  inches  thick  and  webs  \  inch  thick  will  be  somewhere  near  the  required 
section.  For  such  a  section  I  =  104  X  103  (inches)4  units;  while  J=  110  X  103 
units.  From  Table  VI.,  k  for  the  given  ratio  of  depth  to  breadth  is  '413,  so  that 
J'  =  45'5  X  103  (inches)4  units.  Assuming  E  =  30  X  106  Ibs.  per  square  inch  and 
C  =  12  X  106  Ibs.  per  square  inch,  the  effective  value  of  El  :  CJ  becomes  5*73. 

From  Figs.  19  and  20  it  appears  that  the  values  of  the  end  moments  Mn  and  T., 
for  this  value  of  the  ratio  when  </>  =  30,  are  Ma  =  '435  in-2  and  Ta  =  '067  wr2. 

The  effective  load  per  foot  run,  including  the  weight  of  the  girder,  is  approximately 
2*2  tons,  so  that  the  moments  become 


1  See  "  The  Design  of  Plate  Girders  and  Columns,"  Lilley,  or  any  similar  woik. 


GENERAL  PRINCIPLES  OF  DESIGN  OF  THE  BOW-GIRDER     75 

M  =  -435  X  2-2  X  900  =  880  ft.  tons 
T  =  '067  X  2-2  X  900  =  133  ft.  tons 

while  the  shear  force  F  =  2'2  X  30  X  £  X  \  %  =  69  tons. 

—         -LoU 

Flanrfes. — Adopting  a  working  stress  of  6  tons  per  square  inch  in  tension  and  com- 
pression, and  assuming  an  effective  depth  of  57  inches,  we  have 

6  X  af  X  ST  =  88° 

A4: 

.'.  a*  =  61'8  square  inches 

where  af  is  the  flange  area. 

Assuming  this  to  include  |  the  area  of  the  webs  ( =  ^  X  57  =  7  square  inches 
approx.)  the  required  area  of  flange  plates  and  angles  is  54'8  square  inches.  This 
might  be  made  up  of 

2  plates,  f"  X  33"  =  49'5  square  inches 

2  angles  6J"  X  4£"  X  '55"  =  11'5 

Total  61-0 

From  this  is  to  be  deducted  the  area  corresponding  to  two  rivets,  and  assuming  these 
to  require  1-inch  holes,  this  will  be  approximately  5  square  inches,  leaving  an  effective 
area  of  56'0  square  inches,  or  slightly  more  than  is  required. 

Webs. — Calling  aw  the  area  of  the  two  webs,  the  maximum  shear  stress  due  to 

69 
vertical  loading  =  -  -  tons  square  inches.     The  maximum  shear  stress  due  to  torque 

-  #to 

1'547T 

=  — rj—  (p.  65,  equation  48),  where  A  is  the  effective  area  of  the  section  to  resist 
A  f) 

torsion  and  b  is  the  breadth  across  the  webs.     Allowing  ^  inch  between  the  edges  of 
angles  and  of  flange  plates,  26  becomes  equal  to  33  —  10  =  23  inches,  while 
A  =.  (aw  -\-  area  of  a  23"  width  of  flanges) 

99  X  23 

aw  -\        33 

=  aw  -\-  69  square  inches 

The  resultant  shear  stress  in  vertical  and  horizontal  planes  at  the  neutral  axis  is 
then  given  by 

69       1-54  X  133  X  24 
«.        K  +  69)X  23 

Equating  this  to  the  working  shear  stress,  say  3  tons  per  square  inch,  and 
simplifying  gives 

aw2  -  25-8aw  —  1587  =  0, 
from  which  a,0  =  54'4  square  inches. 

If  t  be  the  thickness  of  the  web  plates  this  makes 

It  X  57  =  54-4 

.  • .    t  =  '477  inch 
or,  say,  |  inch. 

Rivets. — Assuming  the  centre  line  of  the  riveting  at  the  junction  of  webs  and 
flanges  to  be  3  inches  from  the  edge  of  the  web,  or  at  a  distance  25'5  inches  from  the 


76  A   STUDY   OF   THE   CIECULAE-AEC   BOW-GIEDEE 

neutral  axis,  the  shear  stress  at  this  point  due  to  the  vertical  loading  is,  by  equation  (56), 
p.  71,  equal  to  0'90  ton  per  square  inch  of  web  section. 

The  shear  stress  at  the  same  point,  due  to  torsion,  is,  by  (47),  p.  65,  equal  to 


1-547- J          M\2I 
~M~  I1    '  (7)   } 


,        «i      25-5 

where  c-=io- 

so  that  this  stress  equals  '2775  X  — 77— 

Ab 

_  -2775  X  1-54  X  133  X  24 

(57  +  69)  X  23 
=  '47  ton  per  square  inch 

The  resultant  horizontal  or  vertical  shear  at  this  point  is  therefore  '90  +  '47  =  T37 
tons  per  square  inch. 

Considering  one  of  the  web  plates,  the  horizontal  shear  force  corresponding  to  the 
shear  stress  over  a  horizontal  length  p  inches  is 

1'37  pt  tons 
=  '685  'p  tons 

Then  if  p  be  the  pitch  of  the  rivets  and  R  the  safe  working  resistance  to  shear  of 
one  rivet 

R 


Adopting  a  working  stress  of  5  tons  per  square  inch  for  rivets  in  shear,  and  using 
£-inch  rivets  (area  "602  square  inch),  gives 


5  X  '602 
J>=  ~-  =  4'4  inches. 


To  allow  for  the  stress  on  the  rivets  due  to  the  tendency  to  distortion  indicated  in 
Fig.  46,  the  pitch  would  be  reduced  to  about  4  inches,  or  alternatively  two  rows  of  rivets 
with  a  correspondingly  greater  pitch  would  be  used. 

Stiffen  ers.  —  Considering  the  web  as  a  column  whose  effective  length  is  \/2 
times  the  distance  between  adjacent  stiffeners  the  allowable  mean  shear  stress  depends 
on  the  ratio  of  this  length  I  to  the  least  radius  of  gyration  "  r  "  of  the  plate.  For  a  J-inch 

plate  r  (  =  —  p=  j  =  -144  and  /  -f-  r  =  6'92£.     In  the  case  in  question  the  mean  stress 

in  the  web  is  approximately  (3  +  T4)-^2  =  2*2  tons,  and  for  this  stress  Moncrieff1 
has  shown  that  the  maximum  permissible  value  of  I  -=-  r  is  about  265.  This  makes 
I  =  265  •—  6*92  =  38'3  inches,  in  which  case  the  distance  between  the  stiffeners  would 
be  38'3  -f-  \/2  —  27  inches.  As  the  shear  diminishes,  this  distance  is  to  be  increased  to 
suit,  up  to  a  maximum  of  about  3  feet  6  inches. 

Over  the  end  bearings  the  stiffeners  should  be  designed  as  columns  of  sufficient 
strength  to  transmit  the  total  load.  Intermediate  stiffeners  would  be  about 
4"  +  3£"  +  f"  angles. 

For  a  more  detailed  examination  of  this  point  and  of  details  of  design  the  reader 
is  advised  to  consult  any  modern  work  on  the  design  of  girders. 

1   J.  M.  Moncrieff,  Trans.  Am.  Soc.  G.  E.,  Vol.  XLV.,  1901.     See  also  "Structural  Engineering," 
Husband  &  Harby,  Longmans  &  Co.,  p.  154. 


TI 


APPENDIX    A 

THE  following  list  of  integrals  will  be  found  of  service  in  solving  the  various 
problems  involved  in  the  circular-arc  bow-girder. 

f  f 

I  0  cos  0  dd  =  6  sin  0  -f  cos  0  ;  0  sin  0  (19  =  sin  0  —  0  cos  0      . 

J 
Q  _  0      sin  20 1  .C        e<w_e__  sin  20 

/*  /* 

cos3  0  tlB  =  sin  0  -  S1^3  -  ;  sin3  dd6=-  C^  (sin2  0-2) 

J 

f*  r'1 

I  sin  (0!  —  0)  d0  =  1  —cos  0i ;  cos  (0X  —  0)  rf0  =  sin  0i 

0 

/»01 

?i  —  0)  d0  =0i  —  sin  0! ;  cos  0 cos  (0i  —  0)  d0  =  ~  cos 0X 

Jo 

c"1 

cos  (0i  —  0)(W  =  \—  cos  0i ;  |  sin  0  cos  (0!  —  0)  d0  = 

cos  0  sin  (0X  —  0)  J0  =  -^~   -1 ; 


2 

0 

V) 


-  - 


cos20sin  (6,  -  d)dd  =\(l+  sin2^  -  cos^)  ;      cos90(0l  -  d)<W  =  ^  (2  cos0!  +  1) 

"  o 

J0  JQ 

r&l  rei 

sin2  0  sin  (^  -  6)  ,W  =         '  C™    l*  ;  sin20cos(01-0)(/0=sin01(l-cos01) 


1- 


rei 

cos3  e  sin  (0!  -  0)  ,W  =  "^  Q  sin  20X  +  |  0 

^o 

f" 
cos3  e  cos  (0i-0)  f/0  =  ~  cos  0j  (sin  2^  -f  20X)  +  f    —  l 

^o 

f  sin3  9  sin  (0,  -  0)  ^  =  J  (sin  ^  +  «°  ^  «"'  ^  _  |  ^  co 


3 

sin3  0  cos  (0!  —  6)  ,18  —  —  sin  0j  (20X  —  sin  2^) 


C&1  rei 

sin  2^  sin  (6l  -  6)  ,W  =  |  sin  ^  (1  -  cos  0j)  ;        sin  20  cos  (0!  -  0)  cW  =  -  ?  cos  2^ 

Jo  J0 


APPENDIX    B. 

MOMENTS    OF    INERTIA    OF    VARIOUS    SECTIONS. 


Section. 


Moments  of  Inertia. 


„„„„ 

m 


m  _.«__ 


TrZ)4 
64 


64 


BD3 
~12" 


64 


64 


12 


12  Z^L>  — 


[D  - 


[BD2  —  M2]2  —  4£Z)kZ  [D  —  d]* 


-  bd2]2  —  4BDbd  [D  -  d]' 
12  [BD  -  bd] 


+ 


12  [BD  -  bd] 

[DB*  -  f/fr2]2  -  4BDbd  [B  -  b]* 
12  [BD  —  bd] 

^  [bD3  +  Bd3~\ 


INDEX 


A. 

Angle  sections,  torsional  rigidity  of,  57 
Appendix  A,  77 
B,  78 

Autenrieth,  investigations  of,  on  the  torsion  of 
beam,  sections  other  than  circular,  59,  72 


B. 

Bach,  researches  of,  63,  69 
Beims,  bending  of,  1 

best  section  to  resist  torsion,  73 
continuous,  3 

having     more     than     two 

supports,  5 

curvature,  deflection,  and  slope  of,  2 
distribution  of  stress  in,  59 
encastre,  effect    of    settlement  of    one 

support,  7,  45 
uniform  loading  of,  3 
unsyinmetrical  loading  of,  8 
with  intermediate  support,  5 
with  no  intermediate  support, 

4 

Box-sections,  distribution  of  shear  stress  in,  70 
torsional  rigidity  of,  56 


C. 

Cantilever,  circular-arc,  with  single  load  at  free 

end, 15 

uniformly  loaded,  16 
straight,  deflection  at  free  end  of,  2 
Castigliano's  theorem,  11 
Channel  sections,  torsional  rigidity  of,  57 
Continuous  beams,  see  Beams. 


D. 

Deflection  of  circular-arc  bow-girder,  14 
cantilever,    2 

straight  beams,  2 

straight  cantilever,  2 
Deflection  produced  by  shear  forces,  13 
Distortion  of  a  beam  section  under  torsion,  74 
Distribution  of  shear  stresses  in  a  beam,  59 


E. 

Effective  polar  moment  of  inertia,  50 

Encastre  beams,  see  Beams. 

Equation  of  three  moments,  6 

Experimental  investigation  of  torsional  rigidity 

of  commercial  sections,  52 
verification      of      formulae      for 

circular- arc  girder,  47 


F. 

Fixing-moments  in  circular-arc  bo  w- girder,  14 
in    encastre   and   continuous 

beams,  3 

Flexual  strength  of  beams,    experimental  in- 
vestigation, 52 
Formulae,   for    deflection    of    bow-girder, 

Chapter  II.,  14 

straight  beams.  2 

for  shear   stress  in   a    beam   under 

torsion,  Autenrieth,  59 
for  torsion  of  beams,  St.  Venant,  72 

G. 

Girder,  box  section,  distribution  of  shear  stress 

in  a,  70 

stiffening  of  a,  74 
circular-arc  bow,  Chapter  II.,  14 

carrying  concentrated 

load,  18 
carrying  uniform  load, 

28 

carrying       uniformly 
loaded  platform,  34 
compound,  46 
effect  of  depression  of 

supports,  45 
equilibrium  of,  14 
general  principles   of 

design  of,  73 
shearing-force  at  any 

section  of  a,  47 
uusymmetrical     load- 
ing, 37 
with         intermediate 

supports,  37 
with       one       central 

support,  37 
with  two  symmetrical 

supports,  40 

semi-circular-arc  bow,  carrying  concen- 
trated load,  24 
semi-circular-arc  bow,  carrying  uniform 

load,  32 

semi-circular-arc  bow,    carrying    uni- 
formly loaded  platform,  34 " 
semi-circular-arc    bow,    supported    by 

cantilever,  45 

semi-circular- arc  bow,  with  two  inter- 
mediate supports,  40 
semi-circular-arc  bow,  with  three  inter- 
mediate supports,  43 
straight,  Chapter  I.,  1 
curvature  of,  2 
deflection  of,  2 

distribution  of  shear  stress  in  a, 
47 


80 


IXDEX 


Girder,  straight,  equilibrium  of,  1 

resilience  of  under  bending,  10 
torsion,  12 

H. 

Hollow  sections,  distribution  of  shear  stress  in, 

65 
effect  on  torsional  rigidity  of 

concrete  filling,  58 
torsional  rigidity  of,  60,  64 
Horizontal  shear  stress  in  a  beam,  47 


I. 


I  sections,  shear  stress  due  to  torque,  68,  TO 

torsional  rigidity  of,  57 
Inertia,  moments  of,  for  various  sections,  78 


J. 


J,  effective  value  of,  in  commercial  sections,  54 


M. 


Maximum  shear  stress  in  a  beam  section  under 

torsion,  69 
Moments,  bending  and  twisting  moments  in  a 

bow-girder,  14 
end  fixing  moments  in  circular-arc 

girder,  14 
of  inertia  of  various  sections,  78 


N. 
Neutral  axis,  shear  stress  at,  59 


P. 


Polar,  moment  of  inertia,  relation  between 
actual  and  theoretical  in  commercial  sections, 
5-1 

Principles  of  design  of  a  bow-girder,  73 


R. 


Relation    between    curvature,    deflection,    and 

slope  of  a  beam,  1 
Resilience,  flexual,  of  beams,  10 

torsional,  of  beams,  11 
Resultant  shear  stress  in  a  beam  subjected  to 

combined  bending  and  twisting,  69 
Rigidity,  torsional,  of  non-circular  sections,  50 

S. 

Sections,  deformation  of,  under  torsion,  74 
moments  of  inertia  for  various,  78 
most  suitable  type,  to  resist  torsion,  73 
Shear  stress  due  to  a  torque,  in  sections  having 

a  continuous  boundary,  60 
due  to  a  torque,  in  hollow  sections, 

65 

in  I  sections,  68,  70 
in  solid  polygonal 

sections,  63 
horizontal,  in  abeam  under  torsion, 

69 
in  a  beam,  due  to  vertical  loading, 

47 
under  combined  loading 

and  torsion,  69 
under  torsion,  69 
in    sections    other    than    circular 

under  torsion,  60 
Shearing  force,  at  any  section  of  a  bow-girder, 

70 

deflection  produced  by,  13 
St.  Venant,  investigations  of,  on  the  torsion  of 

beam  sections  other  than  circular,  51,  52 
Supports,  effect  of  sinking  of  supports,  7,  45 

T. 

Tee  sections,  torsional  rigidity  of,  57 
Theorem,  Castigliano's,  11 

of  three  moments,  6 
Theory  of  bending,  1 
Torsional  rigidity  of  non-circular  sections,  50 

W. 
Webs,  design  of.  in  bow-girder,  75 


BRADBURY,    ACNEW,    &    CO.    Ll>.,    I'KIXTKKS,    LoNlniN    AVIi   TOXI1KIIM  ;K. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
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